Question

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

Explanation:

Step1: Choose substitution

Let $u = x^2 + 1$

Step2: Compute derivative of u

$\frac{du}{dx} = 2x \implies du = 2x dx \implies x dx = \frac{1}{2}du$

Step3: Rewrite integral in terms of u

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

Step4: Integrate with respect to u

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

Step5: Substitute back u

Replace $u$ with $x^2 + 1$ (since $x^2+1>0$, absolute value is unnecessary)

Answer:

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