Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\dpi{110}\int_{0}^{\pi} x.f(\sin x)\; dx=$	A. $x\cdot \int_{0}^{\pi }f \left ( \sin\: x \right )\: dx$
B. $\pi \cdot \int_{0}^{\pi }f \left ( \sin\: x \right )\: dx$
C. $2\pi \cdot \int_{0}^{\pi }f \left ( \sin\: x \right )\: dx$
D. $\frac{\pi}{2} \cdot \int_{0}^{\pi }f \left ( \sin\: x \right )\: dx$
E. $\int_{0}^{\pi }f \left ( \cos\: x \right )\: dx$

$I=\int_{0}^{\pi }\left [x\! \cdot f\left (\sin\: x \right ) \right ]dx \newline Stel \; x=\pi -t \; \Rightarrow \; dx=\!-dt \newline {\color{Yellow} .}\qquad \qquad \qquad \qquad x=0\;\; k.o.m.\; \; t=\pi \newline {\color{Yellow} .}\qquad \qquad \qquad \qquad x=\pi \;\; k.o.m.\; \; t=\0 \newline I=\int_{\pi }^{0} \left [(\pi -t).f(\sin\: t) \right ]\cdot (-dt)\newline = -\pi \int_{\pi }^{0}f(\sin\: t)dt+\int_{\pi }^{0}\left [t\cdot f(sin\: t) \right ]dt\newline =\pi \int_{0}^{\pi }f(\sin\: t)\: dt - \int_{0}^{\pi }\left [ t\cdot f(\sin\: t) \right ]dt\newline \Rightarrow I=\pi \int_{0}^{\pi }f(\sin\: t)\: dt\: -\: I \newline \Rightarrow I=\frac{\pi }{2}\int_{0}^{\pi }f(\sin\: t)\: dt={\color{Red} (!)\: }=\frac{\pi }{2}\int_{0}^{\pi }f(\sin\: x)\: dx$