Evaluate the following :
`int_(-a)^(a)sqrt((a-x)/(a+x))dx`

Explanation

Let `I=int_(-a)^(a)sqrt((a-x)/(a+x))dx`
`=int_(-a)^(a)sqrt(((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`
`=aint_(-a)^(a)(1)/(sqrt(a^(2)-x^(2)))dx-int_(-a)^(a)(x)/(sqrt(a^(2)-x^(2)))dx`
We use the property,
`int_(-a)^(a)f(x)dx=2int_(0)^(a)f(x)dx,` if f is an even function.
= 0, if f is an odd function
Here, `(1)/(sqrt(a^(2)-x^(2)))` is an even function and `(x)/(sqrt(a^(2)-x^(2)))` is an odd function.
`therefore I=2aint_(0)^(a)(1)/(sqrt(a^(2)-x^(2)))dx-0 = 2a[sin^(-1)((x)/(a))]_(-)^(a)`
`=2a[sin^(-1)1-sin^(-1)0]=2a((pi)/(2)-0)=pia.`