Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$\int_{0}^{\frac{\pi}2}\frac{\cos x}{\cos x+\sin x}\;dx=$	A. 1
B.
C.
D. π
E.

$\inline A=\int\frac{\cos x}{\cos x+\sin x}\;dx\qquad B=\int\frac{\sin x}{\cos x+\sin x}\;dx\\A+B=\int\frac{\cos x+\sin x}{\cos x+\sin x}\;dx=\int dx=x+C\\A-B=\int\frac{\cos x-\sin x}{\cos x+\sin x}\;dx=ln\left|\sin x+\cos x\right|+C\\2A=x+ln\left|\sin x+\cos x\right|+C\newline A=\frac{1}{2}x+\frac{1}{2}ln\left|\sin x+\cos x\right|+C\newline\int_{0}^{\frac{\pi}{2}}\frac{\cos x}{\cos x+\sin x}\;dx=\left[\frac{1}{2}x+\frac{1}{2}ln\left|\sin x+\cos x\right|\right]_{0}^{\frac{\pi}{2}}\\{\color{Yellow}.}\quad=\frac{1}{2}\left[\frac{\pi}{2}+\ln\left|1+0\right|-0-\ln\left|0+1\right|\right]=\frac{1}{2}\cdot\frac{\pi}{2}=\frac{\pi}{4}$