3. Determine the primitive $F: \mathbb{R} \longrightarrow \mathbb{R}$ of the function $f: \mathbb{R} \longrightarrow \mathbb{R}$

$$
f(x)=\frac{\sin x \cdot \sin \left(x-\frac{\pi}{4}\right)}{e^{2 x}+\sin ^{2} x}
$$

for which $F(0)=0$.
## Solution.

$$
\begin{aligned}
F(x) & =\frac{\sqrt{2}}{2} \int \frac{\sin ^{2} x-\sin x \cos x}{e^{2 x}+\sin ^{2} x} d x=\frac{\sqrt{2}}{2} \int \frac{e^{2 x}+\sin ^{2} x}{e^{2 x}+\sin ^{2} x} d x-\frac{\sqrt{2}}{4} \int \frac{\left(e^{2 x}+\sin ^{2} x\right)^{\prime}}{e^{2 x}+\sin ^{2} x} d x \\
& =\frac{\sqrt{2}}{2} x-\frac{\sqrt{2}}{4} \ln \left(e^{2 x}+\sin ^{2} x\right)+\mathcal{C}, x \in \mathbb{R}
\end{aligned}
$$

$F(0)=0$ for $\mathcal{C}=0$, so the required primitive is $\left.F(x)=\frac{\sqrt{2}}{2} x-\frac{\sqrt{2}}{4} \ln \left(e^{2 x}+\sin ^{2} x\right), x \in \mathbb{R} . \mathbf{( 1 p}\right)$