Question

Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`

Answer 1

Let I = `int sqrt(x^2 - a^2)(1)dx`   ....(1)

Integrating by parts

I = `sqrt(x^2 - a^2) int 1dx - int(int1dx * d/dx sqrt(x^2 - a^2))dx`

= `xsqrt(x^2 - a^2) - int (x*(2x))/(2sqrt(x^2 - a^2))dx`

= `xsqrt(x^2 - a^2) - int [(x^2 - a^2)/sqrt(x^2 - a^2) + a^2/sqrt(x^2 - a^2)]dx`

= `xsqrt(x^2 - a^2) - int [sqrt(x^2 - a^2) + a^2/sqrt(x^2 - a^2)]dx`

I = `xsqrt(x^2 - a^2) - I - a^2log(x + sqrt(x^2 - a^2))`  ....[From (1)]

∴ 2I = `xsqrt(x^2 - a^2) - a^2log(x + sqrt(x^2 - a^2))`

∴ I = `x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`