Question

1. - / 12.5 points evaluate the indefinite integral. $int\frac{x}{(x^{2}+3)^{2}}dx$ + c

Explanation:

Step1: Use substitution

Let $u = x^{2}+3$, then $du=2x dx$, and $x dx=\frac{1}{2}du$.

Step2: Rewrite the integral

The integral $\int\frac{x}{(x^{2}+3)^{2}}dx$ becomes $\int\frac{1}{2u^{2}}du=\frac{1}{2}\int u^{- 2}du$.

Step3: Integrate using power - rule

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$). For $n=-2$, we have $\frac{1}{2}\int u^{-2}du=\frac{1}{2}\cdot\frac{u^{-2 + 1}}{-2+1}+C$.

Step4: Simplify the result

$\frac{1}{2}\cdot\frac{u^{-1}}{-1}+C=-\frac{1}{2u}+C$.

Step5: Substitute back $u = x^{2}+3$

The final result is $-\frac{1}{2(x^{2}+3)}+C$.

Answer:

$-\frac{1}{2(x^{2}+3)}$