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.

Short answer

The final integral is same.

Step-by-step solution

Given Information

The given integrals are $\int \frac{2x}{x^2+1}dx$ and $\int \frac{1}{x\ln x}dx$.

Change the integral

First integral: $I=\int \frac{2x}{x^2+1}dx$

Using substitution change the integral.

Let $x^2+1=t$

$2xdx=dt$

$I=\frac{dt}{t}$

Second Integral: $I=\int \frac{1}{x\ln x}dx$

Let $\ln x=t$

$\frac{dx}{x}=dt$

$I=\int \frac{dt}{t}$

After the substitution both integral is equal.