29.12. Calculate: a) $\int e^{a x} \sin b x d x$; b) $\int e^{a x} \cos b x d x$.
29.12. a) Integrating by parts, we get

$$
\begin{aligned}
\int e^{a x} \sin b x d x & =-\frac{1}{b} e^{a x} \cos b x+\frac{a}{b} \int e^{a x} \cos b x d x= \\
& =-\frac{1}{b} e^{a x} \cos b x+\frac{a}{b^{2}} e^{a x} \sin b x-\frac{a^{2}}{b^{2}} \int e^{a x} \sin b x d x
\end{aligned}
$$

For \( I=\int e^{a x} \sin b x d x \) we have obtained the relation

$$
I=\frac{e^{a x}}{b^{2}}(-b \cos b x+a \sin b x)-\frac{a^{2}}{b^{2}}(I+C)
$$

Therefore,

$$
\int e^{a x} \sin b x d x=\frac{e^{a x}}{a^{2}+b^{2}}(a \sin b x-b \cos b x)+C
$$

b) Similarly, we get

$$
\int e^{a x} \cos b x d x=\frac{e^{a x}}{a^{2}+b^{2}}(a \cos b x+b \sin b x)+C
$$