integration of (cosnx.sinnx)dx in interval of (0,pie)

To find the integral of \(\cos(nx) \sin(nx)\) over the interval \([0, \pi]\), we use the trigonometric identity for the product of sine and cosine:

\[
\cos(nx) \sin(nx) = \frac{1}{2} \left[ \sin(2nx) \right]
\]

Therefore, the integral becomes:

\[
\int_0^{\pi} \cos(nx) \sin(nx) \, dx = \int_0^{\pi} \frac{1}{2} \sin(2nx) \, dx
\]

Now, let's evaluate this integral:

\[
\int_0^{\pi} \frac{1}{2} \sin(2nx) \, dx = \frac{1}{2} \int_0^{\pi} \sin(2nx) \, dx
\]

The integral of \(\sin(2nx)\) is given by:

\[
\int \sin(2nx) \, dx = -\frac{1}{2n} \cos(2nx)
\]

Substituting this back, we have:

\[
\frac{1}{2} \left[ -\frac{1}{2n} \cos(2nx) \right]_0^{\pi} = -\frac{1}{4n} \left[ \cos(2n\pi) - \cos(0) \right]
\]

Since \(\cos(2n\pi) = 1\) and \(\cos(0) = 1\), the expression simplifies to:

\[
-\frac{1}{4n} \left( 1 - 1 \right) = 0
\]

Hence, the value of the integral is:

\[
\int_0^{\pi} \cos(nx) \sin(nx) \, dx = 0
\]

### Would you like further details or have any questions? Here are some related questions to consider:

1. How can we use different trigonometric identities to simplify integrals?
2. What is the general approach to integrating products of trigonometric functions?
3. How do the limits of integration affect the result when integrating periodic functions?
4. Can the result be different for a different interval, such as \([0, 2\pi]\)?
5. How would the integration change if the argument of the trigonometric functions was different, e.g., \( \cos(mx)\sin(nx) \)?

**Tip:** Always check if there's a trigonometric identity that can simplify the integrand before performing the integration.

integration of (cos(nx) * sin(nx)) dx in interval of (0, π)

Find the integral of cos(nx) * sin(nx) over the interval [0, π] using trigonometric identities. This problem involves the application of integral calculus and trigonometric simplifications, and it's suitable for students in advanced high school or university calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation