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  • If the integral <img alt="\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cint%20%5Cfrac%7B5%20%5Ctan%20x%7D%7B%5Ctan%20x

If the integral \int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k then a is equal to 

Option: 1

-1


Option: 2

-2


Option: 3

1


Option: 4

2


Answers (1)

best_answer

\begin{aligned} \int \frac{5 \tan x}{\tan x-2} & d x=\int \frac{5 \sin x}{\sin x-2 \cos x} d x \\ \\& =\int \frac{(\sin x-2 \cos x)+2(\cos x+2 \sin x)}{(\sin x-2 \cos x)} d x \\ \\& =\int d x+2 \int \frac{\cos x+2 \sin x}{\sin x-2 \cos x} d x \\ \\& =x+2 \ln |(\sin x-2 \cos x)|+k \Rightarrow a=2 \end{aligned}

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