integrate (1 + x^2)^1/2 dx

Question

IntegrateI

Answer 1

Question: integrate (1 + x^2)^1/2 dx

I = int (1 + x^2)^(1/2) dx

let x = sinh(u), then dx = cosh(u) du, and

I = int √(1 + sinh^2(u)) cosh(u) du

I = int cosh^2(u) du

I = int (1/2)(1 + cosh(2u)) du

I = (1/2)(u + (1/2)sinh(2u))

I = (1/2)arcsinh(x) +(1/4)(2.sinh(u).cosh(u))

I = (1/2)arcsinh(x) + (1/2).x.√(1 + sinh^2(u))

I = (1/2)arcsinh(x) + (1/2).x.√(1 + x^2)