Question

find the indefinite integral.

int \frac { x } { 2 + 5 x ^ { 2 } } d x

int \frac { x } { 2 + 5 x ^ { 2 } } d x = square
(use parentheses to clearly denote the argument of each function.)

Explanation:

Step1: Use substitution

Let \(u = 2 + 5x^{2}\), then \(du=10x dx\), and \(x dx=\frac{1}{10}du\).

Step2: Substitute into the integral

\(\int\frac{x}{2 + 5x^{2}}dx=\int\frac{1}{u}\cdot\frac{1}{10}du\).

Step3: Integrate

\(\frac{1}{10}\int\frac{1}{u}du=\frac{1}{10}\ln|u|+C\).

Step4: Back - substitute

Since \(u = 2+5x^{2}\), the integral becomes \(\frac{1}{10}\ln(2 + 5x^{2})+C\).

Answer:

\(\frac{1}{10}\ln(2 + 5x^{2})+C\)