Q.3TB - 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

The final integral is same.

Step 1: Given Information

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

Step 2: 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.