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int sqrt(a^2 - x^2) dx = 1/2 x sqrt (a^2...

` int sqrt(a^2 - x^2) dx = 1/2 x sqrt (a^2 - x^2 ) + 1/2 a^2 sin^-1 (x/a) + c`

Text Solution

Verified by Experts

`"Let I"=int sqrt((a^(2)-x^(2)) )dx=int sqrt(a^(2)-x^(2)). 1dx`
`=sqrt(a^(2)-x^(2)).int 1dx-int [(d)/(dx)(sqrt(a^(2)-x^(2))).int 1dx]dx`
`=sqrt(a^(2)-x^(2)).x-int (-x)/(sqrt(a^(2)-x^(2)))x.dx`
`=x sqrt(a^(2)-x^(3))-int (a^(2)-x^(2)-a^(2))/(sqrt(a^(2)-x^(2)))x`
`=xsqrt(a^(2)-x^(2))-int sqrt(a^(2)-x^(2)dx)+a^(2)int (dx)/(sqrt(a^(2)-x^(2)))`
`=xsqrt(a^(2)-x^(2))-I+a^(2)sin((x)/(a))`
`therefore 2I=xsqrt((a^(2)-x^(2)))+a^(2)sin^(-1)((x)/(a))`
`therefore I=(x)/(2)sqrt((a^(2)-x^(2)))+(a^(2))/(2)sin^(-1)((x)/(a))+c`
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