Math Problem Statement

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

Solution

The integral in the image is:

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

Let's solve it step-by-step.

Step 1: Factor Out the Constant

Since 17\frac{1}{7} is a constant multiplier, we can take it out of the integral:

6+sin(6t)7dt=17(6+sin(6t))dt\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:

17(6+sin(6t))dt=17(6dt+sin(6t)dt)\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 6dt6 \, dt:

    6dt=6t\int 6 \, dt = 6t

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

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

    sin(6t)dt=16cos(6t)\int \sin(6t) \, dt = -\frac{1}{6} \cos(6t)

Step 4: Combine Results

Now, substitute back into the integral:

17(6t16cos(6t))\frac{1}{7} \left( 6t - \frac{1}{6} \cos(6t) \right)

Step 5: Simplify

Distribute the 17\frac{1}{7}:

=6t7142cos(6t)= \frac{6t}{7} - \frac{1}{42} \cos(6t)

Final Answer

6+sin(6t)7dt=6t7142cos(6t)+C\int \frac{6 + \sin(6t)}{7} \, dt = \frac{6t}{7} - \frac{1}{42} \cos(6t) + C

where CC 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 CC when solving indefinite integrals.

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Math Problem Analysis

Mathematical Concepts

Integration
Trigonometric Functions
Constant Multipliers

Formulas

∫ sin(ax) dx = - (1/a) cos(ax)
Constant multiple rule: ∫ c*f(x) dx = c * ∫ f(x) dx

Theorems

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Suitable Grade Level

Grades 11-12 or introductory college-level calculus