https://file.aiquickdraw.com/mathbot/file/1730866523232ns723iqr.jpg

To evaluate the integral

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

we can use the trigonometric identity for \(\cos^2 x\):

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

Thus, we can rewrite the integral as:

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

This simplifies to:

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

Now we can split the integral:

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

1. **First Integral:**

   \[
   \int_0^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2}.
   \]

2. **Second Integral:**

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

   To solve this, let’s find the antiderivative of \(\cos(2x)\):

   \[
   \int \cos(2x) \, dx = \frac{\sin(2x)}{2}.
   \]

   Evaluating this from \(0\) to \(\frac{\pi}{2}\):

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

Therefore, the integral simplifies to:

\[
\int_0^{\frac{\pi}{2}} \cos^2 x \, dx = \frac{1}{2} \left( \frac{\pi}{2} + 0 \right) = \frac{\pi}{4}.
\]

### Final Answer:
\[
\int_0^{\frac{\pi}{2}} \cos^2 x \, dx = \frac{\pi}{4}.
\]

Would you like further details or have any questions?

Here are some related questions to explore:

1. How can we solve integrals involving \(\sin^2 x\) in a similar way?
2. What is the integral of \(\cos^2 x\) over a different interval, like from \(0\) to \(\pi\)?
3. Can we use a different identity to evaluate \(\int \cos^2

Evaluate the integral \( \int_0^{\frac{\pi}{2}} \cos^2 x \, dx \).

Learn how to evaluate the integral \( \int_0^{\pi/2} \cos^2 x \, dx \) by applying trigonometric identities and integration techniques. This problem covers concepts in trigonometry and definite integrals, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation