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
- Use the trigonometric identity for product-to-sum conversion
To simplify the integration, use the identity: $$ ext{cos}(x) ext{sin}(x) = rac{1}{2}[ ext{sin}(2x)] $$
- Substitute the identity into the integral
The integral becomes: $$ ext{∫ cos}(x) ext{sin}(x) , dx = ext{∫} rac{1}{2}[ ext{sin}(2x)] , dx $$
- 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 $dx = rac{1}{2} du$
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Perform the substitution $$ rac{1}{2} ext{∫ sin}(2x) , dx = rac{1}{2} ext{∫ sin}(u) rac{du}{2} = rac{1}{4} ext{∫ sin}(u) , du$$
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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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