Problem 4. Determine the family of primitives

$$
I=\int \frac{x \ln \left(1+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} d x
$$

## NOTE:

Working time 3 hours.

All subjects are mandatory.

Each problem will be graded from 1 to 7 points (1 point is given automatically).

## National Mathematics Olympiad  Local stage, February 16, 2014

## Solutions  Grade 12
Problem 4: Office $1 \mathrm{p}$.
a) $2 \mathrm{p}$.

We make the variable substitution $\sqrt{1+x^{2}}=y$ and thus $\frac{x}{\sqrt{1+x^{2}}} d x=d y$ and

$$
I=\int \ln (1+y) d y
$$

b) $4 \mathrm{p}$.

Using the integration by parts formula, we find

$$
\begin{aligned}
I & =y \ln (1+y)-\int \frac{y}{1+y} d y=y \ln (1+y)-y+\ln (1+y)+\mathcal{C}=(y+1) \ln (1+y)-y+\mathcal{C} \\
& =\left(1+\sqrt{1+x^{2}}\right) \ln \left(1+\sqrt{1+x^{2}}\right)-\sqrt{1+x^{2}}+\mathcal{C}
\end{aligned}
$$