Prove that :
`intsqrt(a^(2)-x^(2))dx=x/2sqrt(a^(2)-x^(2))+(a^(2))/(2)sin^(-1)((x)/(a))+c`

Explanation

Let `I=intsqrt(a^(2)-x^(2))dx`
Intergrating by parts
`I=sqrt(a^(2)-x^(2))int1dx-int[(d)/(dx)(sqrt(a^(2)-x^(2)))*int1dx]dx`
`=xsqrt(a^(2)-x^(2))-int(-2x)/(2sqrt(a^(2)-x^(2)))x*dx`
`=xsqrt(a^(2)-x^(2))-int[sqrt(a^(2)-x^(2))-(a^(2))/(sqrt(a^(2)-x^(2)))dx`
`=x*sqrt(a^(2)-x^(2))-int[sqrt(a^(2)-x^(2))-(a^(2))/(sqrt(a^(2)-x^(2)))]dx`
`=xsqrt(a^(2)-x^(2))-I+a^(2)*sin^(-1)((x)/(a))+c_(1)`
`2I=xsqrt(a^(2)-x^(2))+a^(2)*sin^(-1)((x)/(a))+c_(1)`
`I=x/2sqrt(a^(2)-x^(2))+(a^(2))/(2)*sin^(-1)((x)/(a))+c_(1)`
`therefore intsqrt(a^(2)-x^(2))dx=(x)/(2)sqrt(a^(2)-x^(2))+(a^(2))/(2)sin^(-1)((x)/(a))+c`
where, `c=(c_(1))/(2)*`