Wiskunde Multiple Choice Vraag

De onbepaalde integraal $\int\frac{1}{\sqrt{(1-x^2)^3}}\;dx$ is gelijk aan	A. sin x + C
	B. tan x + C
	C. 1 + x² + C
	D. $-\frac{2}{\sqrt{1-x^2}}+C$
	E. $\frac{x}{\sqrt{1-x^2}}+C$

[ 6-6586 - op net sinds 14.6.2024-(E)- ]

Translation in E N G L I S H

$\int\frac{1}{\sqrt{(1-x^2)^3}}\;dx$ =	A. sin x + C
	B. tan x + C
	C. 1 + x² + C
	D. $-\frac{2}{\sqrt{1-x^2}}+C$
	E. $\frac{x}{\sqrt{1-x^2}}+C$

Oplossing - Solution

Stel x = sin t, dan is 1 − x² = cos² t en is dx = cos t.dt $\\\int\!\!\frac{1}{\sqrt{(1-x^2)^3}}\:dx=\!\int\!\frac{\cos t}{cos^3\,t}\:dt=\!\int\!\frac{1}{\cos^2t}\:dt=\tan t\,+\,C=\frac{\sin t}{\cos t}\,+\,C=\frac{x}{\sqrt{1-x^2}}+C$