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

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

1.	Prove that : `intsqrt(a^(2)-x^(2))dx=x/2sqrt(a^(2)-x^(2))+(a^(2))/(2)sin^(-1)((x)/(a))+c`
Answer» 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)))xdx` `=xsqrt(a^(2)-x^(2))-int[sqrt(a^(2)-x^(2))-(a^(2))/(sqrt(a^(2)-x^(2)))dx` `=xsqrt(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)*`