What is the integration of sin(x) cos(x)?
Understand the Problem
The question is asking for the integral of the product of sine and cosine functions. The approach involves using trigonometric identities or integration techniques to find the antiderivative of sin(x) * cos(x).
Answer
$ -\frac{1}{4} \cos(2x) + C' $
Answer for screen readers
The integral of sin(x) cos(x) is $ -\frac{1}{4} \cos(2x) + C' $
Steps to Solve
- Use a trigonometric identity to simplify the integrand
Use the identity $ \sin(x) \cos(x) = \frac{1}{2} \sin(2x) $ to simplify the integral:
$$ \int \sin(x) \cos(x) , dx = \int \frac{1}{2} \sin(2x) , dx $$
- Simplify the integral
Factor out the constant $ \frac{1}{2} $:
$$ \int \frac{1}{2} \sin(2x) , dx = \frac{1}{2} \int \sin(2x) , dx $$
- Integrate the simplified function
Recall the antiderivative of $ \sin(2x) $:
$$ \int \sin(2x) , dx = -\frac{1}{2} \cos(2x) + C $$
- Combine the results
Combine the constant and the antiderivative:
$$ \frac{1}{2} \left( -\frac{1}{2} \cos(2x) + C \right) = -\frac{1}{4} \cos(2x) + \frac{C}{2} $$
Since any constant, when multiplied by another constant, remains a constant, we can simplify this as:
$$ -\frac{1}{4} \cos(2x) + C' $$
The integral of sin(x) cos(x) is $ -\frac{1}{4} \cos(2x) + C' $
More Information
The structured approach uses trigonometric identities to simplify the integration process. These identities are very useful in calculus for simplifying integrands.
Tips
A common mistake is forgetting to use the correct trigonometric identity, or not properly applying the integration technique after simplifying the integrand.