Q. 90 Use the chain rule and the Funda... [FREE SOLUTION]

Chapter 5: Q. 90 (page 419)

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:
${鈭�}_{a}^{b} f' (u (x)) u' (x) d x = f (u (b)) 鈭� f (u (a))$

Short Answer

Expert verified

After using the chain rule and the Fundamental Theorem of Calculus by the integration-by-substitution formula for definite integrals we proved that ${鈭�}_{a}^{b} f' (u (x)) u' (x) d x = f (u (b)) 鈭� f (u (a))$

Step by step solution

Step 1. Given Information

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:

${鈭�}_{a}^{b} f' (u (x)) u' (x) d x = f (u (b)) 鈭� f (u (a))$

Step 2. To solve taking the left hand side integral.

$y = {鈭�}_{a}^{b} f' (u (x)) u' (x) d x$

Let

$t = u (x) \frac{d t}{d x} = u' (x) d t = u' (x) d x$

Step 3. Now the integral after substitution.

${鈭�}_{a}^{b} f' (u (x)) u' (x) d x = {鈭�}_{a}^{b} f' (t) d t {鈭�}_{a}^{b} f' (u (x)) u' (x) d x = {[f (t)]}_{a}^{b} {鈭�}_{a}^{b} f' (u (x)) u' (x) d x = {[f (u (x))]}_{a}^{b} {鈭�}_{a}^{b} f' (u (x)) u' (x) d x = f (u (b)) - f (u (a))$

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91影视

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:
${鈭�}_{a}^{b} f' (u (x)) u' (x) d x = f (u (b)) 鈭� f (u (a))$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. To solve taking the left hand side integral.

Step 3. Now the integral after substitution.

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Probability and Statistics

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91影视

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:鈭�abf'(u(x))u'(x)dx=f(u(b))鈭�f(u(a))

Short Answer

Step by step solution

Step 1. Given Information

Step 2. To solve taking the left hand side integral.

Step 3. Now the integral after substitution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Applied Mathematics

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:
${鈭�}_{a}^{b} f' (u (x)) u' (x) d x = f (u (b)) 鈭� f (u (a))$