Final answer to the problem
$x\ln\left(\sqrt{x}+\sqrt{1+x}\right)+\frac{1}{2}\ln\left(\sqrt{1+x}+\sqrt{x}\right)-\frac{1}{2}\sqrt{x}\sqrt{1+x}+C_0$
Got another answer? Verify it here!
Step-by-step Solution
Specify the solving method
Choose an option Integrate using basic integrals Integrate by substitution Integrate by parts Integrate using tabular integration Integrate using trigonometric identities Suggest another method or feature
Send
1
We can solve the integral $\int\ln\left(\sqrt{x}+\sqrt{1+x}\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
Intermediate steps
2
First, identify $u$ and calculate $du$
$\begin{matrix}\displaystyle{u=\ln\left(\sqrt{x}+\sqrt{1+x}\right)}\\ \displaystyle{du=\frac{1}{2\sqrt{x}\sqrt{1+x}}dx}\end{matrix}$
Explain this step further
3
Now, identify $dv$ and calculate $v$
$\begin{matrix}\displaystyle{dv=1dx}\\ \displaystyle{\int dv=\int 1dx}\end{matrix}$
5
The integral of a constant is equal to the constant times the integral's variable
$x$
Intermediate steps
6
Now replace the values of $u$, $du$ and $v$ in the last formula
$x\ln\left(\sqrt{x}+\sqrt{1+x}\right)-\int\frac{\sqrt{x}}{2\sqrt{1+x}}dx$
Explain this step further
Intermediate steps
7
The integral $-\int\frac{\sqrt{x}}{2\sqrt{1+x}}dx$ results in: $-\frac{1}{2}\sqrt{x}\sqrt{1+x}+\frac{1}{2}\ln\left(\sqrt{1+x}+\sqrt{x}\right)$
$-\frac{1}{2}\sqrt{x}\sqrt{1+x}+\frac{1}{2}\ln\left(\sqrt{1+x}+\sqrt{x}\right)$
Explain this step further
8
Gather the results of all integrals
$x\ln\left(\sqrt{x}+\sqrt{1+x}\right)+\frac{1}{2}\ln\left(\sqrt{1+x}+\sqrt{x}\right)-\frac{1}{2}\sqrt{x}\sqrt{1+x}$
9
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$x\ln\left(\sqrt{x}+\sqrt{1+x}\right)+\frac{1}{2}\ln\left(\sqrt{1+x}+\sqrt{x}\right)-\frac{1}{2}\sqrt{x}\sqrt{1+x}+C_0$
Final answer to the problem
$x\ln\left(\sqrt{x}+\sqrt{1+x}\right)+\frac{1}{2}\ln\left(\sqrt{1+x}+\sqrt{x}\right)-\frac{1}{2}\sqrt{x}\sqrt{1+x}+C_0$