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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.2
Integrate the functions in Exercises 23 to 59.
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Question : 56 of 271
Marks: +1, -0
tan⁡xsin⁡xcos⁡x\frac{\sqrt{\tan x}}{\sin x \cos x}
Solution:  
Let I=∫tan⁡xsin⁡xcos⁡x dxI = \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx =∫tan⁡xsin⁡xcos⁡x⋅cos⁡2x dx= \int \frac{\sqrt{\tan x}}{\frac{\sin x}{\cos x} \cdot \cos^2 x} \, dx
=∫tan⁡xtan⁡x⋅sec⁡2x dx= \int \frac{\sqrt{\tan x}}{\tan x} \cdot \sec^2 x \, dx
=∫(tan⁡x)−1/2⋅sec⁡2x dx= \int (\tan x)^{-1/2} \cdot \sec^2 x \, dx
Put tan⁡x=t⇒sec⁡2x dx=dt\tan x = t \Rightarrow \sec^2 x \, dx = dt
∴I=∫t−1/2 dt=t−1/2+1−1/2+1+C\therefore I = \int t^{-1/2} \, dt = \frac{t^{-1/2+1}}{-1/2+1} + C =2t1/2+C=2tan⁡x+C= 2t^{1/2} + C = 2\sqrt{\tan x} + C
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