Uppgift 5

(a) $$\begin{aligned}\int\sqrt[7]{6x+5}\,dx&=\int(6x+5)^{1/7}\,dx=\frac16\int t^{1/7}\,dt\\&=\frac16\left(\frac78t^{8/7}\right)+C=\frac7{48}(6x+5)^{8/7}+C,\end{aligned}$$
$t=6x+5$, $dt=6\,dx$, där $C\in\mathbb R$.

(b) Partiell integration:
$$\int_0^bxe^{-2x}\,dx=\left[-\frac12e^{-2x}x\right]_0^b+\frac12\int_0^be^{-2x}\,dx$$
$$u(x)=-\frac12e^{-2x},\quad v(x)=x,\qquad u'(x)=e^{-2x},\quad v'(x)=1.$$
$$\begin{aligned}&=-\frac12be^{-2b}+\frac12\left(-\frac12\right)[e^{-2x}]_0^b\\&=-\frac12be^{-2b}-\frac14(e^{-2b}-1)\to\frac14\quad\text{då }b\to\infty.\end{aligned}$$
$$\implies\int_0^\infty xe^{-2x}\,dx=\frac14.$$