Q. 3 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given Information

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

Step 2. Firstly changing the variable of ∫ 2 x x 2 + 1 d x .

Let $x^2+1=u$

$$\begin{gathered} u=x^2+1 \\ \frac{du}{dx}=2x \\ du=2xdx \end{gathered}$$

This substitution changes the integral into

$\int \frac{2x}{x^2+1}dx=\int \frac{1}{u}du$

Step 3. Now changing the integral ∫ 1 x ln x d x .

Let $\ln x=u$

$$\begin{gathered} u=\ln x \\ \frac{du}{dx}=\frac{1}{x} \\ du=\frac{1}{x}dx \end{gathered}$$

This substitution changes the integral into

$\int \frac{1}{x\ln x}dx=\int \frac{1}{u}du$

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