Question

how to solve this type of integration​

Answer 1

Answer:

$\frac{\pi}{32}$

Step-by-step explanation:

$\int _{0}^{1}\: {x}^{2} (1 - {x}^{2} ) {}^{ \frac{3}{2} } dx$

Put x = sin y

dx = cos y dy

limit 0 to π/2

New integral

$\int _{0}^ \frac{\pi}{2} {} \: \sin {}^{2} (y) (1 - \sin {}^{2} (y) ) {}^{ \frac{3}{2} } \cos(y) dy \\ \int _{0}^ \frac{\pi}{2} {} \: \sin {}^{2} (y) \cos {}^{4} (y) dy$

Which is well known form . using Formula for this

( if you don't know search on internet sin^mx.cos^nx integral )

$= \frac{(1)(3 \times 1)}{6 \times 4 \times 2} \times \frac{\pi}{2} = \frac{\pi}{32}$