If where is a constant of integration

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

Problem

If ∫3cot⁡3x−cot⁡xtan⁡x−3tan⁡3x dx=p f(x)+q g(x)+c\int \frac{3\cot 3x-\cot x}{\tan x-3\tan 3x}\,dx=p\,f(x)+q\,g(x)+c where cc is a constant of integration, then (a) p=1; q=13; f(x)=x; g(x)=ln⁡∣3−tan⁡x3+tan⁡x∣p=1;\ q=\frac{1}{\sqrt{3}};\ f(x)=x;\ g(x)=\ln\left|\frac{\sqrt{3}-\tan x}{\sqrt{3}+\tan x}\right| (b) p=1; q=−13; f(x)=x; g(x)=ln⁡∣3−tan⁡x3+tan⁡x∣p=1;\ q=-\frac{1}{\sqrt{3}};\ f(x)=x;\ g(x)=\ln\left|\frac{\sqrt{3}-\tan x}{\sqrt{3}+\tan x}\right| (c) p=1; q=23; f(x)=x; g(x)=ln⁡∣3+tan⁡x3−tan⁡x∣p=1;\ q=\frac{2}{\sqrt{3}};\ f(x)=x;\ g(x)=\ln\left|\frac{\sqrt{3}+\tan x}{\sqrt{3}-\tan x}\right| (d) p=1; q=−13; f(x)=x; g(x)=ln⁡∣3+tan⁡x3−tan⁡x∣p=1;\ q=-\frac{1}{\sqrt{3}};\ f(x)=x;\ g(x)=\ln\left|\frac{\sqrt{3}+\tan x}{\sqrt{3}-\tan x}\right|

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