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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.3
Find the integrals of the functions in Exercises 62 to 83.
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Question : 76 of 271
Marks: +1, -0
tan⁡32xsec⁡2x\tan^3 2x \sec 2x
Solution:  
Let I=∫tan⁡32xsec⁡2xdxI=\int \tan^3 2x \sec 2x dx
=∫tan⁡22x⋅tan⁡2x⋅sec⁡2xdx=\int \tan^2 2x \cdot \tan 2x \cdot \sec 2x dx
=∫(sec⁡22x−1)⋅sec⁡2x⋅tan⁡2xdx=\int (\sec^2 2x - 1) \cdot \sec 2x \cdot \tan 2x dx
Put sec⁡2x=t\sec 2x = t ⇒2sec⁡2xtan⁡2xdx=dt\Rightarrow 2 \sec 2x \tan 2x dx = dt
∴I=12∫(t2−1)dt\therefore I = \frac{1}{2} \int (t^2 - 1) dt =12(t33−1)+C=\frac{1}{2} \left( \frac{t^3}{3} - 1 \right) + C =16sec⁡22x+C=\frac{1}{6} \sec^2 2x + C
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