Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\dpi{110}\int \left ( \sqrt{\tan x}+\sqrt{\cot x} \right )dx$	A. Bgsin (sin x − cos x) + C
B. Bgsin (cos x − sin x) + C
C. Bgsin (sin x − cos x) + C
D. Bgsin (cos x − sin x)
E. Bgsin (sin x + cos x) + C

see above Bgsin = asin = arcsin	A.
B.
C.
D.
E.

$\begin{aligned} & \int \left ( \sqrt{\tan x} +\sqrt{\cot x}\right )dx \\ & =\int \left ( \frac{\sqrt{\sin x}}{\sqrt{\cos x}}+\frac{\sqrt{\cos x}}{\sqrt{\sin x}} \right )dx\\ & =\int \frac{\sin x+\cos x}{\sqrt{\sin x\cdot \cos x}}dx\\ & =\sqrt{2}\int \frac{\sin x+\cos x}{\sqrt{2\cdot \sin x\cdot \cos x}}dx\\ & =\sqrt{2}\int \frac{\sin x+\cos x}{\sqrt{1-(\sin x - \cos x)^2}}dx\\ & =\sqrt{2}\int \frac{d(\sin x -\cos x)}{\sqrt{1-(\sin x - \cos x)^2}}\\ & {{\color{Yellow} .}}\qquad \qquad^{en\; daar\; \int \frac{d\, f}{\sqrt{1-f^2}}=Bgsin\: f+C}\\ & =\sqrt{2}\cdot Bgsin\,(\sin x-\cos x)+C\end{aligned}$