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Question Number 197436 by horsebrand11 last updated on 17/Sep/23

$$\underset{0}{\stackrel{\pi /2}{\int }}\frac{\mathrm{dx}}{3+\tan \mathrm{x}}=?$$

Answered by Frix last updated on 17/Sep/23

$$\int \frac{dx}{3+\tan x}\stackrel{t=\tan x}{=}\int \frac{dt}{(t+3)(t^2+1)}=$$$$=\frac{1}{10}\int (\frac{1}{t+3}+\frac{3}{t^2+1}-\frac{t}{t^2+1})dt=$$$$=\frac{1}{10}(\ln (t+3)+{3\tan }^{-1}t-\frac{1}{2}\ln (t^2+1))=$$$$=\frac{3x}{10}+\frac{\ln \mid 3\cos x+\sin x\mid }{10}+C$$$$\Rightarrow \mathrm{Answer}\mathrm{is}\frac{3\pi }{20}-\frac{\ln 3}{10}$$

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