To solve the integral

$\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx,$

we can use a trigonometric identity to simplify $\cos^2 x$.

Step 1: Use the Power-Reduction Formula

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

Step 2: Substitute the Identity into the Integral

This transforms the integral into: $\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx = \int_{0}^{\frac{\pi}{2}} \frac{1 + \cos(2x)}{2} \, dx.$

We can factor out the $\frac{1}{2}$: $= \frac{1}{2} \int_{0}^{\frac{\pi}{2}} (1 + \cos(2x)) \, dx.$

Step 3: Split the Integral

Now, split the integral: $= \frac{1}{2} \left( \int_{0}^{\frac{\pi}{2}} 1 \, dx + \int_{0}^{\frac{\pi}{2}} \cos(2x) \, dx \right).$

Step 4: Evaluate Each Integral Separately

1. First Integral: $\int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2}.$

2. Second Integral: For $\int_{0}^{\frac{\pi}{2}} \cos(2x) \, dx$, use substitution. Let $u = 2x$, so $du = 2 \, dx$ or $dx = \frac{du}{2}$.

   Changing the limits:

   • When $x = 0$, $u = 0$.
   • When $x = \frac{\pi}{2}$, $u = \pi$.

   The integral becomes: $\int_{0}^{\frac{\pi}{2}} \cos(2x) \, dx = \int_{0}^{\pi} \cos(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{0}^{\pi} \cos(u) \, du.$

   Now, integrate $\cos(u)$: $\frac{1}{2} \int_{0}^{\pi} \cos(u) \, du = \frac{1}{2} \left[ \sin(u) \right]_{0}^{\pi} = \frac{1}{2} (0 - 0) = 0.$

So, the second integral is zero.

Step 5: Combine Results

The integral now simplifies to: $\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4}.$

Final Answer

$\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx = \frac{\pi}{4}.$

Would you like more details on any specific part of the solution?

Related Questions

1. How do we derive the power-reduction formula for trigonometric functions?
2. Can we solve similar integrals for $\sin^2 x$ over the same interval?
3. What is the significance of using substitution in integrals with trigonometric functions?
4. How would the result change if the limits of integration were different?
5. What other techniques can we use to solve integrals involving $\cos^2 x$?

Tip

For integrals involving squares of trigonometric functions, the power-reduction identities are particularly useful to simplify the expression before integrating.