Q. 20 Consider the integral 鈭�24xx2-1... [FREE SOLUTION]

Chapter 5: Q. 20 (page 417)

Consider the integral ${鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x$ .
(a) Solve this integral by using u-substitution while keeping the limits of integration in terms ofx.
(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms ofu.

Short Answer

Expert verified

(a) The value of integral by using u-substitution while keeping the limits of integration in terms ofxis $\frac{1}{2} [\log 4 - \log 2]$ .

(b) The value of integral again with u-substitution, this time changing the limits of integration to be in terms ofuis $\frac{1}{2} [\log 15 - \log 3]$ .

Step by step solution

Step 1. Given Information

Consider the integral ${鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x$ .

(a) Solve this integral by using u-substitution while keeping the limits of integration in terms of x.

(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Part (a) Step 1. Now solving this integral by using u-substitution while keeping the limits of integration in terms of x.

Let

$u = x^{2} - 1 \frac{d u}{d x} = 2 x d u = 2 x d x \frac{1}{2} d u = x d x$

Part (a) Step 2. This substitution changes the integral into

${鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x = \frac{1}{2} {鈭�}_{2}^{4} \frac{1}{u} d u {鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x = \frac{1}{2} {[\log x]}_{2}^{4} {鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x = \frac{1}{2} [\log 4 - \log 2]$

Part (b) Step 2. Now solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Let

$u = x^{2} - 1 \frac{d u}{d x} = 2 x d u = 2 x d x \frac{1}{2} d u = x d x$

Part (b) Step 2. We will now write the limits of integration in terms of the new variable u

When $x = 2$ we have

role="math" localid="1648803674365" $u = x^{2} - 1 u (2) = {(2)}^{2} - 1 u (2) = 4 - 1 u (2) = 3$

When $x = 4$ we have

$u = x^{2} - 1 u (4) = {(4)}^{2} - 1 u (4) = 16 - 1 u (4) = 15$

Part (a) Step 3. This substitution changes the integral into

${鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x = \frac{1}{2} {鈭�}_{3}^{15} \frac{1}{u} d u {鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x = \frac{1}{2} {[\log x]}_{3}^{15} {鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x = \frac{1}{2} [\log 15 - \log 3]$

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Consider the integral ${鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x$ .
(a) Solve this integral by using u-substitution while keeping the limits of integration in terms ofx.
(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms ofu.

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Now solving this integral by using u-substitution while keeping the limits of integration in terms of x.

Part (a) Step 2. This substitution changes the integral into

Part (b) Step 2. Now solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Part (b) Step 2. We will now write the limits of integration in terms of the new variable u

Part (a) Step 3. This substitution changes the integral into

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91影视

Consider the integral 鈭�24xx2-1dx.(a) Solve this integral by using u-substitution while keeping the limits of integration in terms ofx.(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms ofu.

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Now solving this integral by using u-substitution while keeping the limits of integration in terms of x.

Part (a) Step 2. This substitution changes the integral into

Part (b) Step 2. Now solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Part (b) Step 2. We will now write the limits of integration in terms of the new variable u

Part (a) Step 3. This substitution changes the integral into

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Statistics

Discrete Mathematics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Consider the integral ${鈭�}_{2}^{4} \frac{x}{x^{2} - 1} d x$ .
(a) Solve this integral by using u-substitution while keeping the limits of integration in terms ofx.
(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms ofu.