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

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.11
By using the properties of definite integrals, evaluate the integrals in Exercises 208 to 225.

Question : 210 of 271

Marks: +1, -0

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

Solution:

Let

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

...(i)

I=\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin^{3/2}\left(\frac{\pi}{2}-x\right)}{\sin^{3/2}\left(\frac{\pi}{2}-x\right)+\cos^{3/2}\left(\frac{\pi}{2}-x\right)}dx

\left[ \because \int\limits_{0}^{a} f(x)\,dx = \int\limits_{0}^{a} f(a-x)\,dx \right]

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

..(ii)

Adding (i) and (ii), we have

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

=\int\limits_{0}^{\frac{\pi}{2}}1\,dx = \left[ x \right]_{0}^{\frac{\pi}{2}}

=\frac{\pi}{2}-0 = \frac{\pi}{2}

\Rightarrow I = \frac{\pi}{4}

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