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By using the properties of the definite integral, evaluate the integral: ∫0π2sin32xsin32x+cos32xdx - Mathematics

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Question

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`

Sum
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Solution

Let I = `int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`

I = `int_0^(pi/2) cos^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`

`2I = int_0^(pi/2) (sin^(3/2)x/(sin^(3/2)x+cos^(3/2) x)+cos^(3/2)x/(sin^(3/2)x + cos^(3/2)x)) dx`

Simplify the numerator:

`(sin^(3/2)x+cos^(3/2) x)/(sin^(3/2)x+cos^(3/2)) = 1`

`2I = int_0^(pi/2) 1 dx`

`int_0^(pi/2) 1 dx = [x]_0^(pi/2)=pi/2 - 0 = pi/2`

`2I = pi/2`

`I=pi/4`

`pi/4`

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Chapter 7: Integrals - Exercise 7.11 [Page 347]

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NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 7 Integrals
Exercise 7.11 | Q 3 | Page 347

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