Question

Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`

Answer 1

`int_(-a)^asqrt((a-x)/(a+x)) dx`

Let `I=int_(-a)^asqrt((a-x)/(a+x)) dx`

`=int_(-a)^asqrt(((a-x)(a-x))/((a+x)(a-x))) dx`

`=int_(-a)^a (a-x)/sqrt(a^2-x^2) dx`

`=int_(-a)^a a/sqrt(a^2-x^2) dx-int_(-a)^a x/sqrt(a^2-x^2) dx`

[but `a/sqrt(a^2-x^2)` is an is an even function and `x/sqrt(a^2-x^2)` is an odd function]

`=2a.[sin^-1(x/a)]_0^a`

`=2a.[sin^-1 1-sin^-1 0]`

`=2a[pi/2-0]`

`int_(-a)^asqrt((a-x)/(a+x)).dx=pia`