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4.3- 4.4

4.3- 4.4. Fundamental Theorem of Calculus Indefinite Integrals. Net Area on interval [-1,9] is?. Example: Evaluate A(x). Using geometry:. Using integration:. 4.5. Substitution Rule. Find  x 3 cos ( x 4 + 2) dx . Solution:

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4.3- 4.4

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  1. 4.3- 4.4 Fundamental Theorem of Calculus Indefinite Integrals

  2. Net Area on interval [-1,9] is?

  3. Example: Evaluate A(x)

  4. Using geometry:

  5. Using integration:

  6. 4.5 Substitution Rule

  7. Find x3cos(x4 + 2) dx. Solution: We make the substitution u = x4 + 2 because its differential is du = 4x3 dx, which, apart from the constant factor 4, occurs in the integral. Thus, using x3 dx =du and the Substitution Rule, we have x3cos(x4 + 2) dx = cosudu = cosu du Example:

  8. = sin u + C = sin(x4 + 2)+ C Notice that at the final stage we had to return to the original variable x. Example – Solution cont’d

  9. Evaluate . Solution: Let u = 2x + 1. Then du= 2 dx, so dx = du. So: Example: 4 0

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