Chapter 5: Q. 3 (page 417)

Explain why $\int \frac{2 x}{x^{2} + 1} d x$ and $\int \frac{1}{x \ln x} d x$ are essentially the same integral after a change of variables.

Short Answer

Expert verified

Both integrals turn into $\int \frac{1}{u} d u$ after a change of variables; $u = x^{2} + 1$ in the first case, $u = \ln x$ in the second.

Step by step solution

Step 1. Given Information

Explain why $\int \frac{2 x}{x^{2} + 1} d x$ and role="math" localid="1648740843374" $\int \frac{1}{x \ln x} d x$ are essentially the same integral after a change of variables.

Step 2. Firstly changing the variable of ∫2xx2+1dx.

Let $x^{2} + 1 = u$

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

This substitution changes the integral into

role="math" localid="1648740683170" $\int \frac{2 x}{x^{2} + 1} d x = \int \frac{1}{u} d u$

Step 3. Now changing the integral ∫1xlnxdx.

Let $\ln x = u$

$u = \ln x \frac{d u}{d x} = \frac{1}{x} d u = \frac{1}{x} d x$

This substitution changes the integral into

localid="1648740902640" $\int \frac{1}{x \ln x} d x = \int \frac{1}{u} d u$

Both integrals turn into $\int \frac{1}{u} d u$ after a change of variables; $u = x^{2} + 1$ in the first case, $u = \ln x$ in the second.

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Explain why $\int \frac{2 x}{x^{2} + 1} d x$ and $\int \frac{1}{x \ln x} d x$ are essentially the same integral after a change of variables.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly changing the variable of ∫2xx2+1dx.

Step 3. Now changing the integral ∫1xlnxdx.

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Explain why ∫2xx2+1dxand ∫1xlnxdxare essentially the same integral after a change of variables.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly changing the variable of ∫2xx2+1dx.

Step 3. Now changing the integral ∫1xlnxdx.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Probability and Statistics

Discrete Mathematics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

Explain why $\int \frac{2 x}{x^{2} + 1} d x$ and $\int \frac{1}{x \ln x} d x$ are essentially the same integral after a change of variables.