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The image contains several integral problems. I will begin by solving a selection of them. Let's start with the integral marked as (4) in problem 31:

\[
\int_0^\pi \cos^2 x \, dx
\]

### Solution:
We will use a trigonometric identity to simplify this integral. The identity is:
\[
\cos^2 x = \frac{1 + \cos(2x)}{2}
\]

Thus, the integral becomes:
\[
\int_0^\pi \cos^2 x \, dx = \int_0^\pi \frac{1 + \cos(2x)}{2} \, dx
\]

We can split this into two separate integrals:
\[
\int_0^\pi \frac{1}{2} \, dx + \int_0^\pi \frac{\cos(2x)}{2} \, dx
\]

Now, let's solve these integrals:

1. \(\int_0^\pi \frac{1}{2} \, dx = \frac{1}{2} \times \pi = \frac{\pi}{2}\)
2. \(\int_0^\pi \frac{\cos(2x)}{2} \, dx\)

For the second part, the integral of \(\cos(2x)\) is \(\frac{\sin(2x)}{2}\), so:
\[
\int_0^\pi \frac{\cos(2x)}{2} \, dx = \left[\frac{\sin(2x)}{4}\right]_0^\pi = 0
\]

Thus, the full integral becomes:
\[
\frac{\pi}{2} + 0 = \frac{\pi}{2}
\]

So, the solution to the integral is:

\[
\int_0^\pi \cos^2 x \, dx = \frac{\pi}{2}
\]

Would you like me to solve other integrals from this list or provide further details on this one?

### Follow-up questions:
1. How would the identity change for \(\sin^2 x\), and how would you apply it to solve a similar integral?
2. Can you compute \(\int_0^\pi \cos^2(2x) \, dx\) using the same technique?
3. How would this integral change if the limits were from \(0\) to \(\frac{\pi}{2}\) instead of \(0\) to \(\pi\)?
4. Could you apply this method to solve \(\int_0^{2\pi} \cos^2 x \, dx\)?
5. How does this integral compare to integrals involving \(\cos(2x)\) directly, rather than squared terms?

### Tip:
When solving trigonometric integrals involving squares, always consider using the appropriate double-angle identities to simplify the expression.

Solve the integrals provided in the image, such as ∫_0^π cos^2(x) dx and others.

Explore solutions to integral problems involving trigonometric identities and exponential functions. This page covers integrals such as ∫_0^π cos^2(x) dx, using trigonometric identities and the Fundamental Theorem of Calculus. Suitable for undergraduate students.

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