https://file.aiquickdraw.com/mathbot/file/1724899228913p0u5ioa6.jpg

The equation provided is:

\[
c_0 = \frac{1}{2} \int_{-1}^{1} f(t) \, dt = \frac{1}{2} \left( \int_{-1}^{0} (t+1) \, dt + \int_{0}^{1} (-t+1) \, dt \right)
\]

Let's solve it step by step.

### Step 1: Evaluate the first integral
The first integral is:

\[
\int_{-1}^{0} (t+1) \, dt
\]

To solve this, we find the antiderivative of \( t+1 \):

\[
\int (t+1) \, dt = \frac{t^2}{2} + t + C
\]

Now, evaluate this from \( -1 \) to \( 0 \):

\[
\left[\frac{t^2}{2} + t \right]_{-1}^{0} = \left(\frac{0^2}{2} + 0 \right) - \left(\frac{(-1)^2}{2} + (-1) \right)
\]
\[
= 0 - \left(\frac{1}{2} - 1 \right) = 0 - \left(\frac{1}{2} - 1 \right) = \frac{1}{2}
\]

### Step 2: Evaluate the second integral
The second integral is:

\[
\int_{0}^{1} (-t+1) \, dt
\]

The antiderivative of \( -t+1 \) is:

\[
\int (-t+1) \, dt = -\frac{t^2}{2} + t + C
\]

Now, evaluate this from \( 0 \) to \( 1 \):

\[
\left[-\frac{t^2}{2} + t \right]_{0}^{1} = \left(-\frac{1^2}{2} + 1 \right) - \left(-\frac{0^2}{2} + 0 \right)
\]
\[
= \left(-\frac{1}{2} + 1 \right) - 0 = \frac{1}{2}
\]

### Step 3: Combine the results
Now, combine the results of the two integrals:

\[
c_0 = \frac{1}{2} \left(\frac{1}{2} + \frac{1}{2} \right) = \frac{1}{2} \times 1 = \frac{1}{2}
\]

So, the value of \( c_0 \) is:

\[
c_0 = \frac{1}{2}
\]

Would you like any further details on this calculation?

Here are some related questions to further explore:

1. How would the integral change if the function inside was different, such as \( t^2 \) instead of \( t+1 \)?
2. What happens if the limits of integration change, for example, from \( -2 \) to \( 2 \)?
3. How can you generalize this method to solve similar problems with different piecewise functions?
4. What is the significance of \( c_0 \) in Fourier series, and how is it generally computed?
5. How would the computation change if the function had discontinuities at certain points?

**Tip:** When solving definite integrals, always make sure to compute the antiderivative correctly and carefully apply the limits to avoid errors.

Learn how to solve the definite integral c_0 = (1/2) ∫_{-1}^{1} f(t) dt with a detailed step-by-step solution. This problem covers integral calculus concepts, including the evaluation of definite integrals and antiderivatives using the Fundamental Theorem of Calculus. Suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation