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The subsitution Method

The subsitution Method. Fatema Ahmed Alhajeri 201108382 #19. Previous information. an indefinite integral is a function ∫ f(x)dx= F(x) where F’(x) = f (x) which represents a particular antiderivitive of f, or an entire family of antiderivitives. Example 1. ∫ x √x dx

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The subsitution Method

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  1. The subsitution Method Fatema Ahmed Alhajeri 201108382 #19

  2. Previous information an indefinite integral is a function ∫f(x)dx= F(x) where F’(x) = f(x) which represents a particular antiderivitive of f, or an entire family of antiderivitives.

  3. Example 1 ∫x √x dx = ∫ x x1/2 dx =∫ x (½ + 1) dx = ∫x 3/2dx Therefore using the intermediate integration rule:

  4. We can solve this previous equation: • F(x) = ∫ x3/2 dx • = (x5/2 ) + C 5/2 Therefore, • = F(x) = 2/5 x5/2+C

  5. The Substitution Method We need to use the substitution method in order to convert difficult equations, to fit the intermediate general equations in order to solve. Example 2 f(x) = ∫ 2x √(1+x2) dx = ∫ (1+x2 )1/2 2x dx Using substituion method we let u = 1+x2 since the derivitive of u (du) is = 2x and is present in the equation u = 1 +x2 du= 2x dx

  6. We now change the equation in terms of “u” where it equals to: • F (x)= ∫ (u)1/2 du • This equation fits the structure of: • Thus, • ∫ (u)1/2 du • = ∫ (u)3/2 + C 3/2 • = 2/3 u3/2 + C

  7. Now, replacing the “u” with the variable x to retrieve the final answer: = 2/3 √(1+x2)3/2 + C

  8. Chapter 5INTEGRALS THE SUBSITUTION RULE SomaiaElsherif 5.5 Math prject - Substitution rule

  9. The Substitution rule In general, Notice that each differentiation rule for functions provides a rule to find an antiderivative The substitution method is a rewriting method for integrals where we can extend the scope of these rules The idea behind substitution rule is to replace a relatively complicated integral by a simpler one Math prject - Substitution rule

  10. Summary of the substitution rule proof The Chain rule for differentiation The chain rule implies the substitution rule∫f’(g(x))g’(x) = f(g(x)) + C Substitution rule (according to text book) ∫ f(g(x))g’(x) dx = ∫ f(u) du where u = g(x), then du = g’(x) Math prject - Substitution rule

  11. The Substitution rule If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I,Then ∫ f(g(x))g’(x) dx = ∫ f(u) du Notice that the rule was proved using the chain rule for differentiationThus, It is easy to remember it ! Only you have to think of dx and du in the previous formula as differentials Math prject - Substitution rule

  12. Example Find ∫ x³ cos (x⁴ + 2) dx. We made a substitution u = x⁴ + 2why? Because it’s differentiable function it’s differential is du = 4x³ dx, which, apart from the constant factor 4, occurs in the integral. Thus, using x³ dx = du/4 and the substitution rule we have Math prject - Substitution rule

  13. ∫x³ cos (x⁴ + 2) dx = ∫cos u . ¼ du = ¼ ∫ cos u du = ¼ sin u + C Notice that you have to return to the original variable x = ¼ sin (x⁴ + 2) + C Math prject - Substitution rule

  14. Notice that The main challenge in using the Substitution rule is to think of an appropriate substitution You have 2 ways to do it ! Choose u to be function in the integrand whose differential occurs ( except for a constant factor )this is similar to the previous example case Math prject - Substitution rule

  15. If the first method didn’t work, Try this Choose u to be some how complicated part of the integrand ( perhaps the inner function in a composite function ) Finally ! Finding the right substitution is a bit of art It’s not unusual to guess wrong If your first guess doesn’t work, just try another substitution Math prject - Substitution rule

  16. THE SUBSTITUTION RULE FOR DEFINITE INTEGRALS Done by: MahaMohd.Ibrahim. ID:201106851 Supervised by: Foud Al-muhannadi

  17. General rule If g’ is continuous on [a,b] and f is continuous on the range of u=g(x), then: =(

  18. Example Evaluate: To find the new limit of the integration we note that: When and when

  19. Therefore:

  20. Evaluate: Method (1):

  21. Method (2): ⇒ ⇒ The integral now is:

  22. We got the same answer!

  23. Definite integral :symmetry Gehaddesouky 201104686 #26

  24. Definite integral : symmetry We use the substitution by U to simplify the calculations of integrals of functions that are symmetric Example :

  25. Substitution rule (symmetry) The main rule is : If (F) is continues on [-a,a] if (f) is even then [f(-x)=f(X)], then If (F) is odd then [f(-x)=-f(X)], then

  26. Example 1, Evaluate by writing it As a sum of two integrals and interpreting one of those integrals in terms of an area.

  27. Example 2

  28. Thank you for your time

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