Algebraic Substitution Of Fractions

Question

I already tried to putting the square root like this: $\sqrt{\frac{x}{5 + x}}$ but I dont know what to do next.

$$\int \frac{\sqrt{x}}{\sqrt{5+x}}dx$$

Answer 1

Another method would be to let $u=\sqrt{x+5}$, so $x=u^2-5 \text{ and } dx=2udu$ to get

$\;\;\;\displaystyle\int\frac{\sqrt{u^2-5}}{u}\cdot 2u\;du=2\int{\sqrt{u^2-5}}\;du$ .

Now let $u=\sqrt{5}\sec\theta, du=\sqrt{5}\sec\theta\tan\theta d\theta$ to get $10\int\tan^{2}\theta\sec\theta\;d\theta$, and integrate by parts.

Answer 2

$$\frac x{5+x}=u^2\implies x=\frac{5u^2}{1-u^2}\implies dx=\frac{10u\,du}{(1-u^2)^2}\implies$$

$$\int\sqrt\frac x{5+x}\,dx=\int\frac{10u^2}{(1-u^2)^2}du$$

and now you have the integral of a rational function. Do, for example, partial fractions, or whatever.