## SUBJECT I

Calculate:
a) $\int \sqrt{\frac{e^{x}-1}{e^{x}+1}} d x$
b) $\int \frac{x}{1+x+e^{x}} d x$
## SUBJECT I

Calculate:
a) $\int \sqrt{\frac{e^{x}-1}{e^{x}+1}} d x$
b) $\int \frac{x}{1+x+e^{x}} d x$

Selected by Prof. Paul Florinel from C.T. "Al.Papiu Ilarian" Zalău

(local Botosani 2012)

Grading Rubric

![](https://cdn.mathpix.com/cropped/2024_06_07_2245471222d1192df8feg-2.jpg?height=112&width=1522&top_left_y=1343&top_left_x=198)

$$
=\int \frac{e^{x}}{\sqrt{e^{2 x}-1}} d x-\int \frac{e^{-x}}{\sqrt{1-\left(e^{-x}\right)^{2}}} d x
$$

$$
=\int \frac{\left(e^{x}\right)^{r}}{\sqrt{\left(e^{x}\right)^{2}-1}} d x+\int \frac{\left(e^{-x}\right)^{r}}{\sqrt{1-\left(e^{-x}\right)^{2}}} d x=\ln \left(e^{x}+\sqrt{\left(e^{x}\right)^{2}-1}\right)+\arcsin \left(e^{-x}\right)+C \ldots \ldots . .2 \text{p}
$$

b) using the notation $f(x)=1+x+e^{x}$, we have $x=f(x)-f^{\prime}(x)$................................1p

$$
\int \frac{x}{1+x+e^{x}} d x=\int \frac{f(x)-f^{\prime}(x)}{f(x)} d x=\int d x-\int \frac{f^{\prime}(x)}{f(x)} d x=x-\ln \left(1+x+e^{x}\right) \ldots \ldots \ldots \ldots \ldots \ldots . \ldots \text{p}
$$