Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\dpi{110}\int \frac{x^3+1}{x^4+x^3+x^2+x}\;dx=$	A. ln \| x \| + Bgtan x + C
	B. ln \| x \| − Bgtan x + C
	C. ln \| x + 1 \| + Bgtan x + C
	D. ln \| x + 1 \| − Bgtan x + C
	E. Bgtan x − ln \| x \| + C

[ 6-8712 - op net sinds 3.4.2021-(E)-15.6.2024 ]

Translation in E N G L I S H

see above Bgtan = atan = arctan ln x = log_e x	A.
	B.
	C.
	D.
	E.

Oplossing - Solution

$\int \frac{x^3+1}{x^4+x^3+x^2+x}\: dx \newline =\int \frac{(x+1)(x^2-x+1)}{x^3(x+1)+x(x+1)}\: dx \newline =\int \frac{x^2-x+1}{x^3+x}\: dx \newline =\int \frac{x^2+1}{x(x^2+1)}\: dx\: -\int \frac{x}{x^3+x}\: dx \newline =\int \frac{1}{x}\: dx\: -\int \frac{1}{x^2+1}\: dx \newline =ln\left | x \right |\: -\: Bgtan\: x\; +C$

$https://latex.codecogs.com/png.image?\dpi{110}\int \frac{x^3+1}{x^4+x^3+x^2+x}\;dx=$	A. ln \| x \| + Bgtan x + C
	B. ln \| x \| − Bgtan x + C
	C. ln \| x + 1 \| + Bgtan x + C
	D. ln \| x + 1 \| − Bgtan x + C
	E. Bgtan x − ln \| x \| + C