Integrate via substitution

Question

Need help on solving integrals using subsitution. As I have only solved ones with Newton-Leibniz, I don't know how to solve this types: $$ \int_0^2 \frac{dx}{\sqrt{x+1}+\sqrt{(x+1)^3}} dx$$

Answer 1

Note that $\;x+1>0\;$ :$$\sqrt{x+1}+\sqrt{(x+1)^3}=\sqrt{x+1}\left(\sqrt{(x+1)^2}+1\right)=\sqrt{x+1}(x+2)$$

Substitute now

$$u^2=x+1\implies dx=2u\,du\implies$$

$$\int_0^2\frac{dx}{\sqrt{x+1}(x+2)}=\int_1^{\sqrt3}\frac{2u\,du}{u(u^2+1)}=\left.2\int_1^{\sqrt3}\frac{du}{1+u^2}=2\arctan u\right|_1^{\sqrt3}=$$

$$=2\left(\frac{\pi}3-\frac\pi4\right)=\frac\pi6$$