For most integrals I have came across, u is almost always substituted in the denominator. However, I came across the following integral: $$\int\frac{\sqrt{x}}{1+x}dx$$
I intuitively thought that $1+x$ would be substituted, but according to the solutions, u had to be ${\sqrt{x}}$.
My question is, how do I distinguish between when u has to be substituted in the denominator versus the numerator?
Best Answer
$u=1+x$ is fine. Your integral changes to
$$\int \dfrac{\sqrt{u-1}}{u} du$$
Here you find yourself in need of another substitution though, for example $u=\sec^2t$. Note that this makes $t=\sec^{-1}(\sqrt{x+1})$.
This gives
$$\int \dfrac{\tan t}{\sec^2 t}\cdot 2\sec^2 t\tan t \ dt=2\int \tan^2 t \ dt=2(\tan t-t) + c$$
Back substituting gives the answer
$$2\tan(\sec^{-1}(\sqrt{x+1}))-2\sec^{-1}(\sqrt{x+1})+c$$
This can be simplified further using trigonometric properties if desired.