To evaluate the integral

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

we can simplify it by using a trigonometric identity.

Step 1: Use the Power-Reducing Identity

The identity for $\cos^2(x)$ is: $\cos^2(x) = \frac{1 + \cos(2x)}{2}.$

Step 2: Substitute in the Integral

Substituting $\cos^2(x)$ with $\frac{1 + \cos(2x)}{2}$, we get: $\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1 + \cos(2x)}{2} \, dx.$

Step 3: Separate the Integral

This can be split into two integrals: $\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1}{2} \, dx + \int_0^{\frac{\pi}{2}} \frac{\cos(2x)}{2} \, dx.$

Step 4: Evaluate Each Integral Separately

1. First Integral: $\int_0^{\frac{\pi}{2}} \frac{1}{2} \, dx = \frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4}.$

2. Second Integral: $\int_0^{\frac{\pi}{2}} \frac{\cos(2x)}{2} \, dx = \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx.$ To evaluate $\int \cos(2x) \, dx$, we use the antiderivative: $\int \cos(2x) \, dx = \frac{\sin(2x)}{2}.$ Applying the limits from $0$ to $\frac{\pi}{2}$: $\frac{1}{2} \cdot \left[ \frac{\sin(2x)}{2} \right]_0^{\frac{\pi}{2}} = \frac{1}{2} \cdot \left( \frac{\sin(\pi)}{2} - \frac{\sin(0)}{2} \right) = \frac{1}{2} \cdot (0 - 0) = 0.$

Step 5: Combine Results

Thus, $\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \frac{\pi}{4} + 0 = \frac{\pi}{4}.$

Final Answer

$\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \frac{\pi}{4}.$

Would you like more details or have any questions?

1. How would the result change if the limits of integration were different?
2. How can you apply this power-reducing identity to $\sin^2(x)$?
3. What if the integrand were $\sin^2(x) + \cos^2(x)$?
4. How would you evaluate $\int \cos^2(x) \sin(x) \, dx$?
5. Can you apply this method to find $\int_0^{2\pi} \cos^2(x) \, dx$?

Tip: The power-reducing identities are particularly useful for integrals involving $\sin^2(x)$ or $\cos^2(x)$ over standard intervals, as they simplify the expression into terms that are easier to integrate.