Question

free response questions: show all
) compute $int arctan x dx$.

Explanation:

Step1: Use integration - by - parts formula

The integration - by - parts formula is $\int u\;dv=uv-\int v\;du$. Let $u = \arctan x$ and $dv=dx$. Then $du=\frac{1}{1 + x^{2}}dx$ and $v = x$.

Step2: Apply the formula

$\int\arctan x\;dx=x\arctan x-\int\frac{x}{1 + x^{2}}dx$.

Step3: Solve $\int\frac{x}{1 + x^{2}}dx$

Let $t = 1 + x^{2}$, then $dt = 2x\;dx$ and $\int\frac{x}{1 + x^{2}}dx=\frac{1}{2}\int\frac{dt}{t}=\frac{1}{2}\ln|t|+C=\frac{1}{2}\ln(1 + x^{2})+C$.

Answer:

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