integral of 3cos^2 with bounds 0 and pi/2

We want to evaluate the integral of \( 3 \cos^2(x) \) from \( 0 \) to \( \frac{\pi}{2} \). The integral can be written as:

\[
\int_0^{\frac{\pi}{2}} 3 \cos^2(x) \, dx
\]

We will first use a trigonometric identity to simplify \( \cos^2(x) \). The identity is:

\[
\cos^2(x) = \frac{1 + \cos(2x)}{2}
\]

Substituting this into the integral, we get:

\[
\int_0^{\frac{\pi}{2}} 3 \cdot \frac{1 + \cos(2x)}{2} \, dx
\]

Simplifying:

\[
\frac{3}{2} \int_0^{\frac{\pi}{2}} (1 + \cos(2x)) \, dx
\]

Now, split this into two separate integrals:

\[
\frac{3}{2} \left[ \int_0^{\frac{\pi}{2}} 1 \, dx + \int_0^{\frac{\pi}{2}} \cos(2x) \, dx \right]
\]

### First integral:

\[
\int_0^{\frac{\pi}{2}} 1 \, dx = \frac{\pi}{2}
\]

### Second integral:

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

To solve this, use substitution: Let \( u = 2x \), so \( du = 2 dx \), or \( dx = \frac{du}{2} \).

The limits change as well: when \( x = 0 \), \( u = 0 \); when \( x = \frac{\pi}{2} \), \( u = \pi \).

So the integral becomes:

\[
\frac{1}{2} \int_0^{\pi} \cos(u) \, du
\]

We know that \( \int \cos(u) \, du = \sin(u) \), so:

\[
\frac{1}{2} \left[ \sin(u) \right]_0^{\pi} = \frac{1}{2} \left[ \sin(\pi) - \sin(0) \right] = \frac{1}{2} \cdot (0 - 0) = 0
\]

So, the second integral evaluates to 0.

### Final result:

Now, combining the two results:

\[
\frac{3}{2} \left( \frac{\pi}{2} + 0 \right) = \frac{3}{2} \cdot \frac{\pi}{2} = \frac{3\pi}{4}
\]

Thus, the value of the integral is:

\[
\boxed{\frac{3\pi}{4}}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the integral change if the bounds were different?
2. What other trigonometric identities can simplify similar integrals?
3. How would you integrate \( \cos^4(x) \) over the same bounds?
4. How can substitution help with more complex integrals?
5. What is the significance of definite integrals in calculating areas?

**Tip:** For trigonometric integrals, always check if you can use identities to simplify powers of sine or cosine.

Explore how to compute the definite integral of 3cos^2(x) from 0 to π/2 using trigonometric identities and integration techniques. This solution breaks down the process with key formulas and detailed explanations. Suitable for students in Grades 11-12.

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