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Questions

$If J = \int \frac{1}{\tan^{2} x + \sec^{2} x} d x then J is equal to$

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a

2tan−1(2tanx)−tan−1(2tanx)+K

b

2tan−1(2tanx)−x+K

c

2tan−1(2tanx)+x2+K

d

2tan−1(2tanx)+x+K

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detailed solution

Correct option is B

Let J=∫1tan2⁡x+sec2⁡xdx Let J=∫1tan2⁡x+sec2⁡xsec2⁡xsec2⁡xdx=∫sec2⁡xdx2tan2⁡x+11+tan2⁡x Put tan⁡x=t⇒sec2⁡xdx=dt Let J=∫dt2t2+11+t2=∫22t2+1−11+t2dt=22tan−1⁡(2t)−tan−1⁡(t)+KJ=2tan−1⁡(2tan⁡x)−tan−1⁡(tan⁡x)+KJ=2tan−1⁡(2tan⁡x)−x+K

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