First slide

Evaluation of definite integrals

Question

$\int_{0}^{π / 2} \frac{\sin^{3 / 2} x}{\sin^{3 / 2} x + \cos^{3 / 2} x} dx$ is equal to

Moderate

A

0
B

$\frac{π}{2}$
C

$\frac{π}{4}$
D

None of these

Solution

Let,
$\begin{array}{l} I = \int_{0}^{π / 2} \frac{\sin^{3 / 2} x}{\sin^{3 / 2} x + \cos^{3 / 2} x} dx - - - - (i) \\ then I = \int_{0}^{π / 2} \frac{\sin^{3 / 2} (\frac{π}{2} - x)}{\sin^{3 / 2} (\frac{π}{2} - x) + \cos^{3 / 2} (\frac{π}{2} - x)} dx \\ [∵ \int_{0}^{a} f (x) dx = \int_{0}^{a} f (a - x) dx] \\ \Rightarrow I = \int_{0}^{π / 2} \frac{\cos^{3 / 2} x}{\cos^{3 / 2} x + \sin^{3 / 2} x} dx - - - - - (ii) \end{array}$
$[∵ \sin (\frac{π}{2} - x) = \cos x and \cos (\frac{π}{2} - x) = \sin x]$
On adding Eqs. (i) and (iil, we get
$\begin{array}{l} 2 I = \int_{0}^{π / 2} \frac{\sin^{3 / 2} x + \cos^{3 / 2} x}{\sin^{3 / 2} x + \cos^{3 / 2} x} dx \\ = \int_{0}^{π / 2} 1 dx = [x]_{0}^{π / 2} = \frac{π}{2} - 0 \Rightarrow I = \frac{π}{4} \end{array}$

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