Wiskunde Multiple Choice Vraag

$I=\int_{0}^{+\infty}{\frac{x}{\left(x+1\right)^4}dx}=$	A. \(\frac14\)
B. \(\frac15\)
C. \(\frac16\)
D. \(\frac56\)
E.

$I=\int_{0}^{+\infty}{frac{x}{\left(x+1\right)^4}dx=$	A. $\frac14$
	B. $\frac15$
	C. $\frac16$
	D. $\frac56$
	E. ∞

Stel t = x + 1, dan is dt = dx , de nieuwe grenzen 1 en +oneindig

, en de integraal I
$I=\int_{0}^{+\infty}{\frac{x}{\left(x+1\right)^4}dx}=\int_{1}^{+\infty}{\frac{t-1}{t^4}dt}=\int_{1}^{+\infty}{\left(t^{-3}-t^{-4}\right)dt=\left[\frac{t^{-2}}{-2}-\frac{t^{-3}}{-3}\right]_1^{+\infty}}=-\left(\frac{1}{-2}-\frac{1}{-3}\right)=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$
$=\left[\frac{t^{-2}}{-2}-\frac{t^{-3}}{-3}\right]_1^{+\infty}=-\left(\frac{1}{-2}-\frac{1}{-3}\right)=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$