https://file.aiquickdraw.com/mathbot/file/1730660432281weugq1xu.jpg

The integral in the image is:

\[
\int \frac{6 + \sin(6t)}{7} \, dt
\]

Let's solve it step-by-step.

### Step 1: Factor Out the Constant

Since \(\frac{1}{7}\) is a constant multiplier, we can take it out of the integral:

\[
\int \frac{6 + \sin(6t)}{7} \, dt = \frac{1}{7} \int (6 + \sin(6t)) \, dt
\]

### Step 2: Split the Integral

We can split the integral into two separate integrals:

\[
\frac{1}{7} \int (6 + \sin(6t)) \, dt = \frac{1}{7} \left( \int 6 \, dt + \int \sin(6t) \, dt \right)
\]

### Step 3: Integrate Each Term

1. **Integrate \(6 \, dt\):**

   \[
   \int 6 \, dt = 6t
   \]

2. **Integrate \(\sin(6t) \, dt\):**

   Using the formula \(\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax)\), we get:

   \[
   \int \sin(6t) \, dt = -\frac{1}{6} \cos(6t)
   \]

### Step 4: Combine Results

Now, substitute back into the integral:

\[
\frac{1}{7} \left( 6t - \frac{1}{6} \cos(6t) \right)
\]

### Step 5: Simplify

Distribute the \(\frac{1}{7}\):

\[
= \frac{6t}{7} - \frac{1}{42} \cos(6t)
\]

### Final Answer

\[
\int \frac{6 + \sin(6t)}{7} \, dt = \frac{6t}{7} - \frac{1}{42} \cos(6t) + C
\]

where \(C\) is the constant of integration.

Would you like further clarification on any part of this solution, or details on trigonometric integrals?

Here are some related questions you might find useful:
1. How do we handle integrals with trigonometric functions in general?
2. What is the formula for integrating a constant multiplied by a trigonometric function?
3. How does the constant factor impact the integration process?
4. What other common methods are there for integrating functions like these?
5. How do you verify an indefinite integral solution?

**Tip:** Always remember to include the constant of integration \(C\) when solving indefinite integrals.

Evaluate the integral ∫ (6 + sin(6t)) / 7 dt.

Learn how to evaluate the integral ∫ (6 + sin(6t)) / 7 dt with a detailed, step-by-step solution. This problem covers key calculus concepts, including integration of trigonometric functions and the constant multiple rule, suitable for students in Grades 11-12 or introductory college-level calculus.

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