Integration Techniques

Replace a complex expression with $u$.

$\int 2x(x^2+1)^3 dx$. Let $u = x^2+1$, $du = 2x\,dx$.

$= \int u^3\,du = \frac{u^4}{4} + C = \frac{(x^2+1)^4}{4} + C$.

$\int u\,dv = uv - \int v\,du$

$\int x e^x\,dx$. Let $u = x$, $dv = e^x\,dx$. Then $du = dx$, $v = e^x$.

$= xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1) + C$.

$\int \frac{5x+1}{(x+1)(x-2)}\,dx = \int \frac{2}{x+1} + \frac{3}{x-2}\,dx = 2\ln|x+1| + 3\ln|x-2| + C$.

Use identities: $\sin^2 x = \frac{1-\cos 2x}{2}$, $\cos^2 x = \frac{1+\cos 2x}{2}$.

$\int x\sin x\,dx$: by parts with $u = x$, $dv = \sin x\,dx$.

$= -x\cos x + \int \cos x\,dx = -x\cos x + \sin x + C$.