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2. Using Boundless Presentations Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: • The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. http://boundless.com/teaching-platform • Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: educators@boundless.com Free to share, print, make copies and changes. Get yours at www.boundless.com

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4. Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions Inverse Functions and Advanced Integration Techniques of Integration Further Applications of Integration ] Inverse Functions and Advanced Integration Free to share, print, make copies and changes. Get yours at www.boundless.com

5. Inverse Functions and Advanced Integration > Inverse Functions: Exponential, Logarithmic, an... Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions • Inverse Functions • Derivatives of Exponential Functions • Logarithmic Functions • Derivatives of Logarithmic Functions • The Natural Logarithmic Function: Differentiation and Integration • The Natural Exponential Function: Differentiation and Integration • Exponential Growth and Decay • Inverse Trigonometric Functions: Differentiation and Integration • Hyperbolic Functions • Indeterminate Forms and L'Hôpital's Rule • Bases Other than e and their Applications Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/inverse-functions-and-advanced-integration-3/inverse-functions-exponential-logarithmic-and-trigonometric-functions-13/

6. Inverse Functions and Advanced Integration > Techniques of Integration Techniques of Integration • Basic Integration Principles • Integration By Parts • Trigonometric Integrals • Trigonometric Substitution • The Method of Partial Fractions • Integration Using Tables and Computers • Approximate Integration • Improper Integrals • Numerical Integration Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/inverse-functions-and-advanced-integration-3/techniques-of-integration-14/

7. Inverse Functions and Advanced Integration > Further Applications of Integration Further Applications of Integration • Arc Length and Surface Area • Area of a Surface of Revolution • Physics and Engineering: Fluid Pressure and Force • Physics and Engeineering: Center of Mass • Applications to Economics and Biology • Probability • Taylor Polynomials Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/inverse-functions-and-advanced-integration-3/further-applications-of-integration-15/

8. Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com

9. Inverse Functions and Advanced Integration Key terms • antiderivativean indefinite integral • binarythe bijective base-2 numeral system, which uses only the digits 0 and 1 • cardiovascularRelating to the circulatory system, that is the heart and blood vessels. • centroidthe point at the center of any shape, sometimes called the center of area or the center of volume • curvea simple figure containing no straight portions and no angles • definite integralthe integral of a function between an upper and lower limit • derivativea measure of how a function changes as its input changes • differentiablea function that has a defined derivative (slope) at each point • differentiationthe process of determining the derived function of a function • ethe base of the natural logarithm, • ethe base of the natural logarithm, • ethe base of the natural logarithm, Free to share, print, make copies and changes. Get yours at www.boundless.com

10. Inverse Functions and Advanced Integration • euclidean spaceordinary two- or three-dimensional space (and higher dimensional generalizations), characterized by an infinite extent along each dimension and a constant distance between any pair of parallel lines • exponentthe power to which a number, symbol or expression is to be raised: for example, the in . • exponentialany function that has an exponent as an independent variable • exponentialany function that has an exponent as an independent variable • fluidany substance which can flow with relative ease, tends to assume the shape of its container, and obeys Bernoulli's principle; a liquid, gas, or plasma • fluxthe rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux, magnetic flux • functiona relation in which each element of the domain is associated with exactly one element of the co-domain • Gravitational accelerationacceleration on an object caused by gravity; at different points on Earth, an acceleration between 9.78 and 9.82 m/s2, depending on altitude • indeterminatenot accurately determined or determinable • integralalso sometimes called antiderivative; the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed • integralalso sometimes called antiderivative; the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed • integralalso sometimes called antiderivative; the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed Free to share, print, make copies and changes. Get yours at www.boundless.com

11. Inverse Functions and Advanced Integration • integrandthe function that is to be integrated • integrationthe operation of finding the region in the x-y plane bound by the function • inversea function that undoes another function • inversea function that undoes another function • irrationalof a real number, that cannot be written as the ratio of two integers • irreducibleunable to be factorized into polynomials of lower degree, as • limita value to which a sequence or function converges • linearhaving the form of a line; straight • logarithmthe exponent by which another fixed value, the base, must be raised to produce that number • logarithmthe exponent by which another fixed value, the base, must be raised to produce that number • meromorphicrelating to or being a function of a complex variable that is analytic everywhere in a region except for singularities at each of which infinity is the limit and each of which is contained in a neighborhood where the function is analytic except for the singular point itself • polynomialan expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power Free to share, print, make copies and changes. Get yours at www.boundless.com

12. Inverse Functions and Advanced Integration • polynomialan expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power • Pressurethe amount of force that is applied over a given area divided by the size of this area • probability density functionany function whose integral over a set gives the probability that a random variable has a value in that set • revolutionrotation: the turning of an object around an axis • seriesthe sum of the terms of a sequence • surface areathe total area on the surface of a three-dimensional figure • surplusspecifically, an amount in the public treasury at any time greater than is required for the ordinary purposes of the government • tangenta straight line touching a curve at a single point without crossing it there • torusthe standard representation of such a space in three-dimensional Euclidean space; a shape consisting of a ring with a circular cross-section; the shape of an inner tube or hollow doughnut • transcendentalof or relating to a number that is not the root of any polynomial that has positive degree and rational coefficients • trapezoida (convex) quadrilateral with two (non-adjacent) parallel sides • trapezoidalin the shape of a trapezoid, or having some faces which have one pair of parallel sides Free to share, print, make copies and changes. Get yours at www.boundless.com

13. Inverse Functions and Advanced Integration • trigonometricrelating to the functions used in trigonometry: , , , , , • trigonometricrelating to the functions used in trigonometry: , , , , , • trigonometricrelating to the functions used in trigonometry: , , , , , Free to share, print, make copies and changes. Get yours at www.boundless.com

14. Inverse Functions and Advanced Integration Approximating Deltas For a small piece of curve, can be approximated with the Pythagorean theorem. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Arc length."CC BYhttp://en.wikipedia.org/wiki/Arc_lengthView on Boundless.com

15. Inverse Functions and Advanced Integration Supply and Demand Chart Graph illustrating consumer (red) and producer (blue) surpluses on a supply and demand chart. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Economic surplus."CC BYhttp://en.wikipedia.org/wiki/Economic_surplusView on Boundless.com

16. Inverse Functions and Advanced Integration Blood Flow (a) A tube; (b) The blood flow close to the edge of the tube is slower than that near the center. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Hagen–Poiseuille equation."CC BYhttp://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equationView on Boundless.com

17. Inverse Functions and Advanced Integration Exponential Function as a Taylor Series The exponential function (in blue) and the sum of the first 9 terms of its Taylor series at 0 (in red). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Taylor series."CC BYhttp://en.wikipedia.org/wiki/Taylor_seriesView on Boundless.com

18. Inverse Functions and Advanced Integration Definite Integral A definite integral of a function can be represented as the signed area of the region bounded by its graph. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Integral."CC BYhttp://en.wikipedia.org/wiki/IntegralView on Boundless.com

19. Inverse Functions and Advanced Integration Approximation by Linear Functions The function (in blue) is approximated by a linear function (in red). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Trapezoidal rule."CC BYhttp://en.wikipedia.org/wiki/Trapezoidal_ruleView on Boundless.com

20. Inverse Functions and Advanced Integration Improper Integral of the First Kind The integral may need to be defined on an unbounded domain. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Improper integral."CC BYhttp://en.wikipedia.org/wiki/Improper_integralView on Boundless.com

21. Inverse Functions and Advanced Integration Interactive Graph: Natural Exponential Function The natural exponential function . Notice what happens when you switch the position of the different variables. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless."Interactive Graph:."CC BY-SA 3.0https://www.boundless.com/image/interactive-graph-71ec2fce-6783-4a49-96ca-aec3ab4e6546View on Boundless.com

22. Inverse Functions and Advanced Integration Two Bodies and the COM Two bodies orbiting the COM located inside one body. COM can be defined for both discrete and continuous systems. The two objects are rotating around their center of mass. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Center of mass."CC BYhttp://en.wikipedia.org/wiki/Center_of_massView on Boundless.com

23. Inverse Functions and Advanced Integration Inverse Functions If maps to , then   maps back to . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/9/9d/Inverse_Functions_Domain_and_Range.pngView on Boundless.com

24. Inverse Functions and Advanced Integration Probability Distribution Function Probability distribution function of a normal (or Gaussian) distribution, where mean   and variance . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Normal distribution."CC BYhttps://en.wikipedia.org/wiki/Normal_distributionView on Boundless.com

25. Inverse Functions and Advanced Integration Partial Fraction Expansion The formula for partial fractions is given by f(x)/g(x) Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Partial fraction."Public domainhttps://en.wikipedia.org/wiki/Partial_fractionView on Boundless.com

26. Inverse Functions and Advanced Integration Rectangle Rule Illustration of the rectangle rule. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Numerical integration."CC BYhttp://en.wikipedia.org/wiki/Numerical_integrationView on Boundless.com

27. Inverse Functions and Advanced Integration Integration Numerical integration consists of finding numerical approximations for the value . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Numerical integration."CC BYhttp://en.wikipedia.org/wiki/Numerical_integrationView on Boundless.com

28. Inverse Functions and Advanced Integration Exponential Function The natural exponential function . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Exp.png."CC BY-SA 4.0https://en.wikipedia.org/wiki/Exponential_functionView on Boundless.com

29. Inverse Functions and Advanced Integration Exponential Growth This graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Exponential.png."CC BY-SA 3.0https://en.wikipedia.org/wiki/Exponential_growthView on Boundless.com

30. Inverse Functions and Advanced Integration Arctangent and Arccotangent The usual principal values of the  and functions graphed on the Cartesian plane. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Arctangent_Arccotangent.svg."CC BY-SA 4.0https://en.wikipedia.org/wiki/Inverse_trigonometric_functionsView on Boundless.com

31. Inverse Functions and Advanced Integration Arcsecant and Arccosecant Principal values of the and functions graphed on the Cartesian plane. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Arcsecant_Arccosecant.svg."CC BY-SA 4.0https://en.wikipedia.org/wiki/Inverse_trigonometric_functionsView on Boundless.com

32. Inverse Functions and Advanced Integration Graph of an Exponential Function Graph of the exponential function illustrating that its derivative is equal to the value of the function. From any point  on the curve (blue), let a tangent line (red), and a vertical line (green) with height  be drawn, forming a right triangle with a base  on the -axis. Since the slope of the red tangent line (the derivative) at  is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function,  must be equal to the ratio of  to . Therefore, the base  must always be . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Exp_tangent.svg/321px-Exp_tangent.svg.pngView on Boundless.com

33. Inverse Functions and Advanced Integration Improper Integral of the Second Kind An improper Riemann integral of the second kind.The integral may fail to exist because of a vertical asymptote in the function. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Improper integral."CC BYhttp://en.wikipedia.org/wiki/Improper_integralView on Boundless.com

34. Inverse Functions and Advanced Integration Taylor Series Approximations for The Taylor polynomials for only provide accurate approximations in the range . Note that, for , the Taylor polynomials of higher degree are worse approximations. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."LogTay.png."CC BY-SA 4.0https://en.wikipedia.org/wiki/Natural_logarithmView on Boundless.com

35. Inverse Functions and Advanced Integration Interactive Graph: Taylor Polynomial The Taylor polynomials for only provide accurate approximations in the range −1 < x ≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless."Interactive Graph: Taylor Polynomial."CC BY-SA 3.0https://www.boundless.com/image/interactive-graph-taylor-polynomialView on Boundless.com

36. Inverse Functions and Advanced Integration Trapezoidal Rule Illustration of the trapezoidal rule. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Numerical integration."CC BYhttp://en.wikipedia.org/wiki/Numerical_integrationView on Boundless.com

37. Inverse Functions and Advanced Integration Surface of Revolution A portion of the curve rotated around the -axis (vertical in the figure). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Surface of revolution."CC BYhttp://en.wikipedia.org/wiki/Surface_of_revolutionView on Boundless.com

38. Inverse Functions and Advanced Integration Interactive Graph: Exponential Growth The graph illustrates how exponential growth (purple) surpasses both linear (red) and cubic (blue) growth. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless."Interactive Graph:."CC BY-SA 3.0https://www.boundless.com/image/interactive-graph-80c46671-1581-407e-99ce-95bc905f8897View on Boundless.com

39. Inverse Functions and Advanced Integration Plot of The graph of the logarithm to base 2 crosses the x-axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Binary_logarithm_plot_with_ticks.png."CC BY-SA 4.0https://en.wikipedia.org/wiki/LogarithmView on Boundless.com

40. Inverse Functions and Advanced Integration Interactive Graph: Graph of Log Base 2 The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 2^3 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless."Interactive Graph:."CC BY-SA 3.0https://www.boundless.com/image/interactive-graph-46fcf0f9-9377-4d84-931f-e7b0cc7e9c5eView on Boundless.com

41. Inverse Functions and Advanced Integration A Function and its Inverse A function and its inverse, . Because maps to , the inverse  maps back to . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/c/c8/Inverse_Function.pngView on Boundless.com

42. Inverse Functions and Advanced Integration Fluid Pressure and Force Pressure as exerted by particle collisions inside a closed container. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Pressure."CC BYhttp://en.wikipedia.org/wiki/PressureView on Boundless.com

43. Inverse Functions and Advanced Integration Interactive Graph: Graph of the Logarithmic Function The graph of the logarithm to base 2, . It crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3 because 23 = 8. The graph approaches the y axis but does not meet or intersect it. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless."Interactive Graph:."CC BY-SA 3.0https://www.boundless.com/image/interactive-graph-35a3b586-cb04-4d04-93e6-d641361a6248View on Boundless.com

44. Inverse Functions and Advanced Integration Integration By Parts Integration by parts may be thought of as deriving the area of the blue region from the total area and that of the red region. The area of the blue region is . Similarly, the area of the red region is . The total area, , is equal to the area of the bigger rectangle, , minus the area of the smaller one, : . Assuming the curve is smooth within a neighborhood, this generalizes to indefinite integrals , which can be rearranged to the form of the theorem: . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Integration by parts."CC BYhttp://en.wikipedia.org/wiki/Integration_by_partsView on Boundless.com

45. Inverse Functions and Advanced Integration Interactive Graph: Inverse Trig Functions of arctan and arccot The usual principal values of the arctan (red) and arccot (blue) functions graphed on the Cartesian plane. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless."Interactive Graph: Inverse Trig Functions."CC BY-SA 3.0https://www.boundless.com/image/interactive-graph-inverse-trig-functions-2a3ea3fc-164d-4c86-9094-fe3b90816e13View on Boundless.com

46. Inverse Functions and Advanced Integration Arcsine and Arccosine The usual principal values of the and functions graphed on the Cartesian plane. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Arcsine_Arccosine.svg."CC BY-SA 4.0https://en.wikipedia.org/wiki/Inverse_trigonometric_functionsView on Boundless.com

47. Inverse Functions and Advanced Integration Interactive Graph: Inverse Trig Functions of arcsin and arccos The usual principal values of the arcsin (blue) and arccos (red) functions graphed on the Cartesian plane. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless."Interactive Graph: Inverse Trig Functions."CC BY-SA 3.0https://www.boundless.com/image/interactive-graph-inverse-trig-functionsView on Boundless.com

48. Inverse Functions and Advanced Integration Attribution • Wiktionary."inverse."CC BY-SA 3.0http://en.wiktionary.org/wiki/inverse • Wiktionary."function."CC BY-SA 3.0http://en.wiktionary.org/wiki/function • Wikipedia."Inverse function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_function • Wiktionary."tangent."CC BY-SA 3.0http://en.wiktionary.org/wiki/tangent • Wiktionary."e."CC BY-SA 3.0http://en.wiktionary.org/wiki/e • Wiktionary."exponential."CC BY-SA 3.0http://en.wiktionary.org/wiki/exponential • Wikipedia."Exponential function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Exponential_function • Wiktionary."exponent."CC BY-SA 3.0http://en.wiktionary.org/wiki/exponent • Wiktionary."binary."CC BY-SA 3.0http://en.wiktionary.org/wiki/binary • Wikipedia."Logarithmic function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithmic_function • Wikipedia."Logarithmic function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithmic_function • Wikipedia."logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm • Wiktionary."e."CC BY-SA 3.0http://en.wiktionary.org/wiki/e • Wikipedia."Logarithmic functions."CC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithmic_functions • Wikibooks."Calculus/Derivatives of Exponential and Logarithm Functions."CC BY-SA 3.0http://en.wikibooks.org/wiki/Calculus/Derivatives_of_Exponential_and_Logarithm_Functions • Wiktionary."transcendental."CC BY-SA 3.0http://en.wiktionary.org/wiki/transcendental • Wiktionary."irrational."CC BY-SA 3.0http://en.wiktionary.org/wiki/irrational Free to share, print, make copies and changes. Get yours at www.boundless.com

49. Inverse Functions and Advanced Integration • Wikipedia."Natural logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/Natural_logarithm • Wikipedia."Natural logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/Natural_logarithm • Wikipedia."Natural logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/Natural_logarithm • Boundless Learning."Boundless."CC BY-SA 3.0http://www.boundless.com//management/definition/differentiation • Wiktionary."e."CC BY-SA 3.0http://en.wiktionary.org/wiki/e • Wikibooks."Calculus/Derivatives of Exponential and Logarithm Functions."CC BY-SA 3.0http://en.wikibooks.org/wiki/Calculus/Derivatives_of_Exponential_and_Logarithm_Functions • Wiktionary."polynomial."CC BY-SA 3.0http://en.wiktionary.org/wiki/polynomial • Wiktionary."linear."CC BY-SA 3.0http://en.wiktionary.org/wiki/linear • Wiktionary."exponential."CC BY-SA 3.0http://en.wiktionary.org/wiki/exponential • Wikipedia."Exponential growth."CC BY-SA 3.0http://en.wikipedia.org/wiki/Exponential_growth • Wikipedia."Exponential growth."CC BY-SA 3.0http://en.wikipedia.org/wiki/Exponential_growth • Wiktionary."trigonometric."CC BY-SA 3.0http://en.wiktionary.org/wiki/trigonometric • Wikipedia."List of integrals of inverse trigonometric functions."CC BY-SA 3.0http://en.wikipedia.org/wiki/List_of_integrals_of_inverse_trigonometric_functions • Wikipedia."Differentiation of trigonometric functions."CC BY-SA 3.0http://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions • Wikipedia."Inverse trigonometric functions."CC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_trigonometric_functions • Wiktionary."inverse."CC BY-SA 3.0http://en.wiktionary.org/wiki/inverse • Wiktionary."meromorphic."CC BY-SA 3.0http://en.wiktionary.org/wiki/meromorphic • Wikipedia."Hyperbolic function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Hyperbolic_function • Wiktionary."limit."CC BY-SA 3.0http://en.wiktionary.org/wiki/limit Free to share, print, make copies and changes. Get yours at www.boundless.com

50. Inverse Functions and Advanced Integration • Wiktionary."indeterminate."CC BY-SA 3.0http://en.wiktionary.org/wiki/indeterminate • Wiktionary."differentiable."CC BY-SA 3.0http://en.wiktionary.org/wiki/differentiable • Wikipedia."L'Hôpital's rule."CC BY-SA 3.0http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule • Wikipedia."Indeterminate form."CC BY-SA 3.0http://en.wikipedia.org/wiki/Indeterminate_form • Wikipedia."logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm • Wikipedia."Logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm • Wikipedia."Binary logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/Binary_logarithm • Wiktionary."integration."CC BY-SA 3.0http://en.wiktionary.org/wiki/integration • Wikipedia."Integral."CC BY-SA 3.0http://en.wikipedia.org/wiki/Integral • Wiktionary."integral."CC BY-SA 3.0http://en.wiktionary.org/wiki/integral • Wikipedia."derivative."CC BY-SA 3.0http://en.wikipedia.org/wiki/derivative • Wikipedia."Integration by parts."CC BY-SA 3.0http://en.wikipedia.org/wiki/Integration_by_parts • Wiktionary."integral."CC BY-SA 3.0http://en.wiktionary.org/wiki/integral • Wiktionary."trigonometric."CC BY-SA 3.0http://en.wiktionary.org/wiki/trigonometric • Wikipedia."List of integrals of trigonometric functions."CC BY-SA 3.0http://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions • Wiktionary."trigonometric."CC BY-SA 3.0http://en.wiktionary.org/wiki/trigonometric • Wikipedia."Trigonometric substitution."CC BY-SA 3.0http://en.wikipedia.org/wiki/Trigonometric_substitution • Wiktionary."irreducible."CC BY-SA 3.0http://en.wiktionary.org/wiki/irreducible • Wikipedia."Partial fractions in integration."CC BY-SA 3.0http://en.wikipedia.org/wiki/Partial_fractions_in_integration Free to share, print, make copies and changes. Get yours at www.boundless.com