By using the properties of definite integrals, evaluate the integrals`int_0^(pi/2)(sin^(3/2)x dx)/(sin^(3/2)+cos^(3/2)x)`

Explanation

Solution :

I=`int_(π/2)^(0) (sin^(3/2)x)/((sin^(3/2)x)(cos^(3/2)x))` -(1)

:`I=int_a^(b)f(x)dx=int_a^(b)f(a+b-x)dx`

I=`int_(0)^(π/2) (sin^(3/2)(π/2-x))/((sin^(3/2)(π/2-x))(cos^(3/2)(π/2-x)))`

I=`int_(π/2)^(0) (cos^(3/2))/((cos^(3/2)x)(sin^(3/2)x))` -(2)

Adding 1 and 2, we get,

`2I=int_(π/2)^(0) ((sin^(3/2))(cos^(3/2)x))/((sin^(3/2)x)(cos^(3/2)x))`

=`int_(π/2)^(0) 1.dx`

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