To solve the integral $\int_{0}^{\pi/4} \sin(2x) \, dx$, we will follow these steps:

Step-by-Step Solution

1. Identify the Integral: $\int \sin(2x) \, dx$

2. Substitution: To solve this integral, use a substitution method. Let $u = 2x$. Then, $du = 2 \, dx$ or $dx = \frac{1}{2} \, du$.

3. Change Limits of Integration: When $x = 0$, $u = 2 \times 0 = 0$.

   When $x = \frac{\pi}{4}$, $u = 2 \times \frac{\pi}{4} = \frac{\pi}{2}$.

   The integral becomes: $\int_{0}^{\pi/4} \sin(2x) \, dx = \frac{1}{2} \int_{0}^{\pi/2} \sin(u) \, du$

4. Evaluate the Integral: $\frac{1}{2} \int_{0}^{\pi/2} \sin(u) \, du = \frac{1}{2} [-\cos(u)]_{0}^{\pi/2}$

5. Calculate the Definite Integral: $= \frac{1}{2} \left( -\cos\left(\frac{\pi}{2}\right) + \cos(0) \right)$

   $= \frac{1}{2} \left( -0 + 1 \right) = \frac{1}{2} \times 1 = \frac{1}{2}$

Final Answer

$\int_{0}^{\pi/4} \sin(2x) \, dx = \frac{1}{2}$

Let me know if you have any questions or would like further clarification on any part of this process.

Related Questions

1. How do you solve integrals using substitution?
2. What is the geometric interpretation of definite integrals?
3. How do the limits of integration change during substitution?
4. What are some common trigonometric identities used in integration?
5. Can you solve $\int \cos(2x) \, dx$ using a similar method?
6. What is the difference between definite and indefinite integrals?
7. How can you verify the result of a definite integral?
8. What other integration techniques are there besides substitution?

Tip

When using substitution in integrals, always remember to change the limits of integration to match the new variable to simplify the process.