TITLE: Integral of fractional expression $\int^3_0 \frac{dx}{1+\sqrt{x+1}}$
QUESTION [5 upvotes]: I want to solve this integral and think about call $\sqrt{x+1} = t \rightarrow t^2 = x+1$
$$\int^3_0 \frac{dx}{1+\sqrt{x+1}}$$
Now the integral is : $$\int^3_0 \frac{2tdt}{1+t}$$ now I need your suggestions.

Thanks.

REPLY [2 votes]: $$\begin{align}
\int^2_0 \frac{2t\, dt}{1+t} &= 2\int^2_0 \frac{1+t-1}{1+t}\,dt \\
&= 2\int^2_0 \left( \frac{1+t}{1+t}-\frac{1}{1+t}\right)\,dt \\
&= 2\int^2_0\left( 1-\frac{1}{1+t}\right)\,dt \\
&= 2[ t-\log({1+t})]_0^2 \\
&= 2[ 2-\log 3] \\
\end{align}$$