Evaluate the indefinite integral $\int \frac{ \sqrt{x+1} }{x} \,\mathrm dx$

Question

I can solve derivatives up to second order and started to learn integration recently, and I have learnt the basic formulae and method of substitutions.

I'm now stuck in this problem: $$\int \frac{ \sqrt{x+1} }{x} \,\mathrm dx.$$

Answer 1

We continue from OP's work in the comments, wherein the substitution $$x = t^2 - 1, \qquad dx = 2 t \,dt$$ transforms the integral to $$2 \int\frac{t^2 \,dt}{t^2 - 1}.$$

Hint We can rearrange our integrand as $$2 \int\left(1 - \frac{1}{1 - t^2}\right)dt .$$

Integrating each term gives $$2(t - \operatorname{artanh} t) + C .$$

Alternatively, we can apply the method of partial fractions to write the integral in $t$ as $$\int \left(2 - \frac{1}{t + 1} + \frac{1}{t - 1}\right) dt .$$