Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\dpi{110}\int_{0}^{\frac{\pi}2} \frac{\cos x}{\cos x + \sin x}\; dx=$	A. 1
B.
C.
D. π
E.

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