Question

evaluate the indefinite integral \\(\int \frac{x}{x^2 + 1} dx\\) \\(\boxed{}\\) + c

Explanation:

Step1: Choose substitution

Let $u = x^2 + 1$, then $du = 2x dx \Rightarrow x dx = \frac{1}{2}du$.

Step2: Substitute into integral

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

Step3: Integrate $\frac{1}{u}$

$\frac{1}{2}\ln|u| + C$.

Step4: Substitute back $u$

$\frac{1}{2}\ln(x^2 + 1) + C$ (since $x^2 +1 >0$, absolute value is unnecessary).

Answer:

$\frac{1}{2}\ln(x^2 + 1)$