# Integrate cos(x) sin(x)

#### Understand the Problem

The question is asking for the integral of the product of cos(x) and sin(x). This involves finding the antiderivative of the function cos(x) * sin(x).

#### Answer

-\frac{1}{4} \text{cos}(2x) + C
##### Answer for screen readers

The final answer is: -\frac{1}{4} \text{cos}(2x) + C

#### Steps to Solve

1. Use the trigonometric identity for product-to-sum conversion

To simplify the integration, use the identity: $$ext{cos}(x) ext{sin}(x) = rac{1}{2}[ ext{sin}(2x)]$$

1. Substitute the identity into the integral

The integral becomes: $$ext{∫ cos}(x) ext{sin}(x) , dx = ext{∫} rac{1}{2}[ ext{sin}(2x)] , dx$$

1. Integrate the function

Factor out the constant and integrate: $$ext{∫} rac{1}{2}[ ext{sin}(2x)] , dx = rac{1}{2} ext{∫ sin}(2x) , dx$$ Use substitution where $u = 2x$, thus $du = 2dx$ which means $dx = rac{1}{2} du$

1. Perform the substitution $$rac{1}{2} ext{∫ sin}(2x) , dx = rac{1}{2} ext{∫ sin}(u) rac{du}{2} = rac{1}{4} ext{∫ sin}(u) , du$$

2. Integrate with respect to $u$ The integral of $sin(u)$ is $-cos(u)$ $$rac{1}{4} [- ext{cos}(u)] + C = - rac{1}{4} ext{cos}(2x) + C$$

Therefore, the final answer is:

The final answer is: -\frac{1}{4} \text{cos}(2x) + C

#### More Information

This type of problem is a good exercise in using trigonometric identities and substitution in integration.

#### Tips

A common mistake is to forget to apply the substitution correctly. Ensure that you substitute $dx$ correctly with $du$. Also, remember to re-substitute $u = 2x$ back into your final answer.

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