I'm trying to perform the following integration:
$$ \int \dfrac{x}{1+\sqrt{x}}\ dx $$
so I marked $t = 1 + \sqrt{x}$ and $dt = x + (2/3)x^{3/2} + C$
I got stuck here after trying to simplify it more or try to break it into two different integrals.
How can I move forward here?
Best Answer
We can play around with the substitution choice to ensure that the integral is expressed purely in terms of $t$.
From the substitution choice we infer: $$t=\sqrt{x}+1$$ $$\implies t-1=\sqrt{x}$$ $$\implies (t-1)^2=x.$$
As GEdgar said, after using the substitution $t=\sqrt{x}+1$, you should obtain: $$dt=\frac{1}{2\sqrt{x}}dx$$ $$\implies 2\sqrt{x}dt=2(t-1)=dx.$$
We now make use of these equalities back in the integral:
$$\int \frac{x}{\sqrt{x}+1}dx$$ $$=\int \frac{2(t-1)^2(t-1)}{t}dt$$ $$=2\int \frac{(t-1)^3}{t}dt.$$
We can expand the numerator and then divide each term by $t$, like so:
$$2\int\frac{t^3-3t^2+3t-1}{t}dt$$ $$=2\int t^2-3t+3-\frac{1}{t}dt$$
Can you proceed from here?