Chapter 5: Q. 89 (page 419)

Use the chain rule to prove the formula for integration by substitution:
$\int f' (u (x)) u' (x) d x = f (u (x)) + C .$

Short Answer

Expert verified

After using the chain rule for integration by substitution its is proved that $\int f' (u (x)) u' (x) d x = f (u (x)) + C .$

Step by step solution

Step 1. Given Information

Use the chain rule to prove the formula for integration by substitution:

$\int f' (u (x)) u' (x) d x = f (u (x)) + C .$

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

$y = \int f' (u (x)) u' (x) d x$

Let

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

Step 3. Now the integral after substitution.

$\int f' (u (x)) u' (x) d x = \int f' (t) d t \int f' (u (x)) u' (x) d x = f (t) + C$

Substituting the value of $t$ .

$\int f' (u (x)) u' (x) d x = f (u (x)) + C$

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Use the chain rule to prove the formula for integration by substitution:
$\int f' (u (x)) u' (x) d x = f (u (x)) + C .$

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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Use the chain rule to prove the formula for integration by substitution:∫f'(u(x))u'(x)dx=f(u(x))+C.

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

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Use the chain rule to prove the formula for integration by substitution:
$\int f' (u (x)) u' (x) d x = f (u (x)) + C .$