₀^{π/2}³x\,dx=2/3 - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Prove the following (Exercises 262 to 267)

Question : 265 of 271

Marks: +1, -0

\int\limits_{0}^{\pi/2}\sin^3x\,dx=\frac{2}{3}

Solution:

\int\limits_{0}^{\pi/2}\sin^3x\,dx

=\frac{1}{4}\int\limits_{0}^{\pi/2}(3\sin x-\sin 3x)\,dx

=\frac{1}{4}\left[-3\cos x+\frac{\cos 3x}{3}\right]^{\pi/2}_{0}

=\frac{1}{4}\left[-3\cos\frac{\pi}{2}+\frac{1}{3}\cos\frac{3\pi}{2}\right]-\frac{1}{4}\left[-\cos 0+\frac{\cos 0}{3}\right]

=\frac{1}{4}\left[0+0+3-\frac{1}{3}\right]

=\frac{1}{4}\left(\frac{8}{3}\right)=\frac{2}{3}

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