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Integrals

Integrals . Keity Okazaki Emily Wang Lisa Wang. Antiderivative. A function is an antiderivative of on if for all

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Integrals

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  1. Integrals Keity Okazaki Emily Wang Lisa Wang

  2. Antiderivative • A function is an antiderivative of on if for all • If is the antiderivative of on then every other antiderivative on is of the form for some constant C.

  3. Indefinite Integral • Indefinite Integral: means that The indefinite integral is the same as the general antiderivative.

  4. Definite Integral • The definite integral of over is the limit of Riemann sums and is denoted by the integral sign: When this limit exists, we say that is integrable over

  5. Integration Techniques • Power Rule for Integrals for • Antiderivative of for

  6. Integration Techniques • Substitution Method If , then • Change of variables formula for definite integrals: If is differentiable on and is integrable on the range of , then

  7. Integration Techniques Cont. • Basic Trigonometric Integrals • For constants:

  8. Integration Techniques • Integrals involving inverse trig functions with positive constant

  9. Integration Techniques Cont. • Integrals involving • For any constant and • For integrals involving

  10. Integration Techniques Cont. • Integration by Parts u= Inverse trig Logarithmic Algebraic Trig dv=Exponential

  11. Integration Techniques Cont. • Trigonometric Integrals-combination of substitution and integration by parts to integrate trigonometric functions Consider the integral Where are positive integers.

  12. Trigonometric Integrals continued • Example 1: Odd power of Evaluate Then use substitution with and

  13. Trigonometric Integrals Continued • Odd Power of or Evaluate

  14. Integration Techniques Cont. • Partial Fractions Suppose that a rational function is proper, in other words, the degree of is less than the degree of , and that the denominator factors as a product of distinct linear factors: Where the roots are all distinct and deg Then there is a partial fraction decomposition: Once the partial fraction decomposition has been found, each term can individually be integrated.

  15. Integration Techniques Cont. • Improper Integrals Fix a number and assume that is integrable over for all The improper integral of over is defined as the limit When the limit exists, we say that the improper integral converges.

  16. Riemann sums • Trapezoidal Rule: The Nth trapezoidal approximation to Where and

  17. Approximations Continued • Midpoint Rule The Nth Midpoint Approximation to is where and is the midpoint of the jth interval

  18. Approximations continued • Right Hand The Right Hand Approximation looks specifically at the y-value from the right at , and multiplies together,

  19. Approximations Cont. • Left hand The Left Hand Approximation is like the Right Hand Approximation, but it looks at the left side for . Keep in mind the y-value will be smaller than from the right hand sum.

  20. Approximations Cont. Right hand rule Left hand rule

  21. Fundamental Theorem of Calculus If is continuous from and is an antiderivative of , then

  22. Fundamental Theorem of Calculus, Part II • Let be a continuous function on . Then is an antiderivative of , so Or, equivalently

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