Uppgift 4

(a) Partiell integration med $u(x)=-\frac15\cos(5x)$ och $v(x)=x$ ger $$\int_0^\pi x\sin(5x)\,dx=\left[-\frac15\cos(5x)\,x\right]_0^\pi-\int_0^\pi\left(-\frac15\cos(5x)\right)\,dx=\frac\pi5+\frac1{25}\left[\sin(5x)\right]_0^\pi=\frac\pi5.$$

(b) Substitution $t=\arctan x$, $dt=\frac1{1+x^2}dx$, ger $$\int\frac{1}{(x^2+1)\arctan x}\,dx=\int\frac1t\,dt=\ln\lvert t\rvert+C=\ln\lvert\arctan x\rvert+C.$$