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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 : 66 of 271
Marks: +1, -0
sin⁡3xcos⁡3x\sin^3 x \cos^3 x
Solution:  
Let I=∫sin⁡3xcos⁡3x dxI=\int \sin^3 x \cos^3 x \, dx =∫sin⁡x⋅sin⁡2x⋅cos⁡3x dx=\int \sin x \cdot \sin^2 x \cdot \cos^3 x \, dx
=∫sin⁡x(1−cos⁡2x)cos⁡3x dx=\int \sin x (1-\cos^2 x) \cos^3 x \, dx =∫(cos⁡3x−cos⁡5x)sin⁡x dx=\int (\cos^3 x-\cos^5 x) \sin x \, dx
Put cos⁡x=t\cos x = t ⇒−sin⁡x dx=dt\Rightarrow -\sin x \, dx = dt
∴I=−∫(t3−t5) dt\therefore I=-\int (t^3-t^5) \, dt =−t44+t66+C=-\frac{t^4}{4}+\frac{t^6}{6}+C =16cos⁡6x−14cos⁡4x+C=\frac{1}{6}\cos^6 x-\frac{1}{4}\cos^4 x+C
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