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

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-2

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2

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-1

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detailed solution

Correct option is C

I=∫5tan⁡xtan⁡x−2dx=∫5sin⁡xsin−2cos⁡xdxLet 5sin⁡x=A(sin⁡x−2cos⁡x)+B(cos⁡x+2sin⁡x) Equating coefficients of sin⁡x and cos⁡x, we get 5=A+2B and 0=−2A+B⇒ A=1,B=2∴ I=∫(sin⁡x−2cos⁡x)+2(cos⁡x+2sin⁡x)sin⁡x−2cos⁡xdx =x+2log⁡|sin⁡x−2cos⁡x|+K∴ a=2

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