Question: Evaluate the integral.
$$\int e^{7\theta} \sin(8\theta) \, d\theta$$
1. **State the problem:** We want to evaluate the integral $$\int e^{7\theta} \sin(8\theta) \, d\theta$$.
2. **Formula and method:** We use integration by parts, which states:
$$\int u \, dv = uv - \int v \, du$$
3. **Choose parts:** Let
$$u = \sin(8\theta)$$
$$dv = e^{7\theta} d\theta$$
4. **Compute derivatives and integrals:**
$$du = 8 \cos(8\theta) d\theta$$
$$v = \frac{1}{7} e^{7\theta}$$
5. **Apply integration by parts:**
$$\int e^{7\theta} \sin(8\theta) d\theta = u v - \int v \, du = \sin(8\theta) \cdot \frac{1}{7} e^{7\theta} - \int \frac{1}{7} e^{7\theta} \cdot 8 \cos(8\theta) d\theta$$
6. **Simplify:**
$$= \frac{1}{7} e^{7\theta} \sin(8\theta) - \frac{8}{7} \int e^{7\theta} \cos(8\theta) d\theta$$
7. **Evaluate the second integral using integration by parts again:**
Let
$$I = \int e^{7\theta} \sin(8\theta) d\theta$$
$$J = \int e^{7\theta} \cos(8\theta) d\theta$$
For $$J$$, let
$$u = \cos(8\theta), \quad dv = e^{7\theta} d\theta$$
Then
$$du = -8 \sin(8\theta) d\theta, \quad v = \frac{1}{7} e^{7\theta}$$
So
$$J = \cos(8\theta) \cdot \frac{1}{7} e^{7\theta} - \int \frac{1}{7} e^{7\theta} (-8 \sin(8\theta)) d\theta = \frac{1}{7} e^{7\theta} \cos(8\theta) + \frac{8}{7} I$$
8. **Substitute back:**
$$I = \frac{1}{7} e^{7\theta} \sin(8\theta) - \frac{8}{7} J = \frac{1}{7} e^{7\theta} \sin(8\theta) - \frac{8}{7} \left( \frac{1}{7} e^{7\theta} \cos(8\theta) + \frac{8}{7} I \right)$$
9. **Simplify and solve for $$I$$:**
$$I = \frac{1}{7} e^{7\theta} \sin(8\theta) - \frac{8}{49} e^{7\theta} \cos(8\theta) - \frac{64}{49} I$$
Add $$\frac{64}{49} I$$ to both sides:
$$I + \frac{64}{49} I = \frac{1}{7} e^{7\theta} \sin(8\theta) - \frac{8}{49} e^{7\theta} \cos(8\theta)$$
$$\frac{113}{49} I = \frac{1}{7} e^{7\theta} \sin(8\theta) - \frac{8}{49} e^{7\theta} \cos(8\theta)$$
Divide both sides by $$\frac{113}{49}$$:
$$I = \left( \frac{1}{7} e^{7\theta} \sin(8\theta) - \frac{8}{49} e^{7\theta} \cos(8\theta) \right) \cdot \frac{49}{113}$$
10. **Final answer:**
$$\boxed{\int e^{7\theta} \sin(8\theta) d\theta = \frac{7}{113} e^{7\theta} \sin(8\theta) - \frac{8}{113} e^{7\theta} \cos(8\theta) + C}$$
where $$C$$ is the constant of integration.