Stuck on an integration question…

Question

$$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$ The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$ but I keep going wrong. Can someone show me how to get this solution?

Thanks.

Answer 1

The form of the answer suggests integration by parts with the choice $$u = \cosh^{-1} \left( \frac{x}{2} + 1 \right), \quad dv = x^{-1/2} \, dx.$$ Then compute the derivative $$du = \ldots?$$ and the integral $$v = \ldots?$$ If you have trouble computing $du$, you can obtain it by writing $$x = \cosh u, \quad \frac{dx}{du} = \sinh u,$$ hence $$\frac{du}{dx} = \frac{1}{\sinh u} = \ldots.$$