antiderivative of sin(2x)cos(2x)
Understand the Problem
The question is asking for the antiderivative (or integral) of the function sin(2x) * cos(2x). To solve this, we can use trigonometric identities or substitution techniques to simplify the expression before integrating.
Answer
$$ -\frac{1}{8} \cos(4x) + C $$
Answer for screen readers
The antiderivative of $\sin(2x) \cos(2x)$ is: $$ -\frac{1}{8} \cos(4x) + C $$
Steps to Solve
- Utilize a Trigonometric Identity
We can use the double angle identity for sine, which states: $$ \sin(2A) = 2 \sin(A) \cos(A) $$ In this case, we rewrite the function: $$ \sin(2x) \cos(2x) = \frac{1}{2} \sin(4x) $$ This simplifies our expression considerably.
- Set Up the Integral
Now we can rewrite our integral: $$ \int \sin(2x) \cos(2x) , dx $$ As: $$ \int \frac{1}{2} \sin(4x) , dx $$
- Integrate the Function
Next, we integrate: $$ \int \frac{1}{2} \sin(4x) , dx $$ Using the fact that the integral of $\sin(kx)$ is: $$ -\frac{1}{k} \cos(kx) + C $$ We apply this to our integral: $$ \frac{1}{2} \left( -\frac{1}{4} \cos(4x) \right) + C $$
- Simplify the Result
Now we simplify: $$ -\frac{1}{8} \cos(4x) + C $$
The antiderivative of $\sin(2x) \cos(2x)$ is: $$ -\frac{1}{8} \cos(4x) + C $$
More Information
Integrating the product of sine and cosine commonly involves recognizing identities that can simplify the expression. Double angle identities are particularly useful in these scenarios.
Tips
- Forgetting to apply the double-angle identity correctly.
- Not simplifying the integral before attempting to integrate.
- Making sign errors during integration, especially with trigonometric functions.