integral from -pi/2 to pi/2(3 sin(8x)  + 700 cos(81x) ) dx  =

We are tasked with evaluating the definite integral:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( 3 \sin(8x) + 700 \cos(81x) \right) dx
\]

This integral can be split into two separate integrals:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 3 \sin(8x) \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 700 \cos(81x) \, dx
\]

### 1. Integral of \( 3 \sin(8x) \)
Since sine is an odd function, i.e., \( \sin(-x) = -\sin(x) \), the integral of \( \sin(kx) \) over a symmetric interval \( [-a, a] \) is zero:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin(8x) \, dx = 0
\]

Thus, the first integral becomes:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 3 \sin(8x) \, dx = 3 \times 0 = 0
\]

### 2. Integral of \( 700 \cos(81x) \)
Since cosine is an even function, i.e., \( \cos(-x) = \cos(x) \), the integral of \( \cos(kx) \) over a symmetric interval \( [-a, a] \) is twice the integral from \( 0 \) to \( a \):

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(81x) \, dx = 2 \int_0^{\frac{\pi}{2}} \cos(81x) \, dx
\]

The integral of \( \cos(kx) \) is \( \frac{\sin(kx)}{k} \), so:

\[
\int_0^{\frac{\pi}{2}} \cos(81x) \, dx = \left[ \frac{\sin(81x)}{81} \right]_0^{\frac{\pi}{2}} = \frac{\sin\left(81 \cdot \frac{\pi}{2}\right)}{81} - \frac{\sin(0)}{81}
\]

Now, \( \sin\left( 81 \cdot \frac{\pi}{2} \right) = \sin\left( \frac{81\pi}{2} \right) = \sin\left( 40\pi + \frac{\pi}{2} \right) = 1 \), since \( \sin\left( n\pi + \frac{\pi}{2} \right) = 1 \) for any integer \( n \).

Thus, the integral becomes:

\[
\int_0^{\frac{\pi}{2}} \cos(81x) \, dx = \frac{1}{81}
\]

Now, multiplying by 2 to account for the symmetric interval:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(81x) \, dx = 2 \times \frac{1}{81} = \frac{2}{81}
\]

Finally, multiplying by 700:

\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 700 \cos(81x) \, dx = 700 \times \frac{2}{81} = \frac{1400}{81}
\]

### Final Answer:
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( 3 \sin(8x) + 700 \cos(81x) \right) dx = \frac{1400}{81}
\]

Let me know if you need more details or have any questions!

Here are some related questions:
1. What is the significance of odd and even functions in integrals?
2. How do you compute the integral of general trigonometric functions over symmetric intervals?
3. What is the role of periodicity in sine and cosine integrals?
4. How can you verify the value of \( \sin\left( \frac{81\pi}{2} \right) \)?
5. How does changing the limits of integration affect the result of trigonometric integrals?

**Tip**: Recognizing the symmetry of functions (odd/even) can simplify many definite integrals, especially over symmetric intervals.

integral from -pi/2 to pi/2(3 sin(8x) + 700 cos(81x)) dx =

Learn how to evaluate the definite integral from -π/2 to π/2 of the function 3 sin(8x) + 700 cos(81x). This problem covers key concepts in definite integrals, properties of odd and even functions, and integration techniques. Suitable for undergraduate students.

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