The antiderivative (or integral) of \( \sin(mx) \) with respect to \(x\) is not \( m \cos(mx) + C \). The correct form is:

\[
\int \sin(mx) \, dx = -\frac{1}{m} \cos(mx) + C
\]

### Explanation:
- To integrate \( \sin(mx) \), we use the substitution method:
  1. Let \( u = mx \), so \( du = m \, dx \), or \( dx = \frac{du}{m} \).
  2. Substitute into the integral:
  \[
  \int \sin(mx) \, dx = \int \sin(u) \cdot \frac{du}{m} = \frac{1}{m} \int \sin(u) \, du
  \]
  3. The integral of \( \sin(u) \) is \( -\cos(u) \), so:
  \[
  \frac{1}{m} \cdot (-\cos(u)) + C = -\frac{1}{m} \cos(mx) + C
  \]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions:
1. What is the antiderivative of \( \cos(mx) \)?
2. How does substitution work in integration?
3. What is the general formula for integrating trigonometric functions?
4. What happens when \( m = 1 \) in this integral?
5. How does integration by parts differ from substitution?

**Tip**: Always check your integral by differentiating the result to see if you get the original function.

Understand the antiderivative of sin(mx) and learn the correct formula with a step-by-step explanation using the substitution method. This solution includes the key integration formula and covers core calculus concepts. 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