29.3. Compute $\int \frac{d x}{x\left(1+\ln ^{2} x\right)} ; \int \frac{e^{x} d x}{1+e^{2 x}} ; \int \frac{\cos x d x}{1+\sin ^{2} x}$.
29.3. According to problems 29.2 and 28.12

\[
\int \frac{d x}{x\left(1+\ln ^{2} x\right)}=\int \frac{d \ln x}{1+\ln ^{2} x}=\int \frac{d y}{1+y^{2}}=\operatorname{arctg} y+C=\operatorname{arctg} \ln x+C ;
\]

\[
\int \frac{e^{x} d x}{1+e^{2 x}}=\int \frac{d e^{x}}{1+e^{2 x}}=\int \frac{d y}{1+y^{2}}=\operatorname{arctg} y+C=\operatorname{arctg} e^{x}+C ;
\]

\[
\int \frac{\cos x d x}{1+\sin ^{2} x}=\int \frac{d \sin x}{1+\sin ^{2} x}=\int \frac{d y}{1+y^{2}}=\operatorname{arctg} y+C=\operatorname{arctg} \sin x+C.
\]