## SUBJECT 1

Let $f: R \rightarrow R, f(x)=\frac{1}{e^{2 x}+1}$.

a) Calculate $\int e^{x} f(x) d x$.

b) Determine the primitive F of $f$ with $F(0)=1$.

c) Calculate $\int(f(x)+f(-x)) d x$.

GMB
## SUBJECT 1

a) With the change of variable $t=e^{x}$ we obtain $\int e^{x} f(x) d x=\operatorname{arctg}\left(e^{x}\right)+C$

b) With the change of variable $t=e^{x}$ and $\int \frac{1}{e^{2 x}+1} d x=\int \frac{e^{x}}{e^{3 x}+e^{x}} d x$, we calculate

$\int \frac{d t}{t^{3}+t}=\ln (t)-\frac{1}{2} \ln \left(t^{2}+1\right)+C$, returning to the substitution we have $\int \frac{1}{e^{2 x}+1} d x=\ln \left(e^{x}\right)-\frac{1}{2} \ln \left(e^{2 x}+1\right)+C$, since $F(0)=1$ we have $c=1+\ln \sqrt{2}$ ..3p. We used that $\frac{1}{t^{3}+t}=\frac{1}{t}-\frac{t}{t^{2}+1}$.

c) We have successively $\int\left(\frac{1}{e^{2 x}+1}+\frac{1}{e^{-2 x}+1}\right) d x=\int\left(\frac{1}{e^{2 x}+1}+\frac{e^{2 x}}{e^{2 x}+1}\right) d x=\int 1 d x=x+C$