∫[1/2 (cos(5πx) + cos(3πx))] dx
= -1/2 [sin(5πx)/5π + sin(3πx)/3] + C

Q: ∫[1/2 (cos(5πx) + cos(3πx))] dx = -1/2 [sin(5πx)/5π + sin(3πx)/3] + C

The image shows the steps of integrating the function $\frac{1}{2} [\cos(5\pi x) + \cos(3\pi x)]$. Here's a breakdown of the integration process: 1. **Expression to Integrate:** \[ \int \frac{1}{2} [\cos(5\pi x) + \cos(3\pi x)] \, dx \] 2. **Separate the Integrals:** \[ = \frac{1}{2} \left[ \int \cos(5\pi x) \, dx + \int \cos(3\pi x) \, dx \right] \] 3. **Integrate Each Term:** - The integral of $\cos(5\pi x)$ is $\frac{\sin(5\pi x)}{5\pi}$. - The integral of $\cos(3\pi x)$ is $\frac{\sin(3\pi x)}{3\pi}$. 4. **Combine the Results:** \[ = \frac{1}{2} \left[ \frac{\sin(5\pi x)}{5\pi} + \frac{\sin(3\pi x)}{3\pi} \right] + C \] Where $C$ is the constant of integration.

Question

Accepted Answer

The image shows the steps of integrating the function $\frac{1}{2} [\cos(5\pi x) + \cos(3\pi x)]$.

Here's a breakdown of the integration process:

1. **Expression to Integrate:**

$$
   \int \frac{1}{2} [\cos(5\pi x) + \cos(3\pi x)] \, dx
   $$

2. **Separate the Integrals:**

$$
   = \frac{1}{2} \left[ \int \cos(5\pi x) \, dx + \int \cos(3\pi x) \, dx \right]
   $$

3. **Integrate Each Term:**

- The integral of $\cos(5\pi x)$ is $\frac{\sin(5\pi x)}{5\pi}$.
   - The integral of $\cos(3\pi x)$ is $\frac{\sin(3\pi x)}{3\pi}$.

4. **Combine the Results:**

$$
   = \frac{1}{2} \left[ \frac{\sin(5\pi x)}{5\pi} + \frac{\sin(3\pi x)}{3\pi} \right] + C
   $$

Where $C$ is the constant of integration.