Integrating sin(2x)dx: Solving for the Integral of a Trigonometric Function

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In summary, the integral of sin(2x)dx can be solved using substitution, with u = 2x and du = 2dx. This method is the reverse of the chain rule in differentiation. By substituting u = 2x, the correct answer of -1/2(cos(2x)) +C can be obtained.
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Homework Statement


[tex]\int[/tex] sin(2x)dx

Homework Equations


I know the integral of sin(x)dx = -cos(x) + C

The Attempt at a Solution


What I did was to say that the integral is -cos(2x) +C, which isn't the correct answer...I should have gotten -1/2(cos(2x)) +C. I can see that this is the correct answer when I differentiate it via chain rule and get sin(2x), however I can't seem to integrate the problem to get the right answer. Can someone walk me through it please...
 
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Use substitution, with u = 2x, du = 2dx. Integration by substitution is the reverse of the chain rule in differentiation.
 
  • #3
ah so that's where I went wrong...I tried u substitution but I used u=sin2x instead of u=2x. thanks mark44.
 

1. What is the integral of sin(2x)dx?

The integral of sin(2x)dx is (-1/2)cos(2x) + C, where C is the constant of integration.

2. How do you solve the integral of sin(2x)dx?

To solve the integral of sin(2x)dx, you can use the power rule for integration, which states that the integral of x^n is (x^(n+1))/(n+1) + C. In this case, n = -1/2, so the integral becomes (sin(2x))^(1/2)/(1/2) + C. Simplifying this further gives (-1/2)cos(2x) + C.

3. Can you explain the steps to find the integral of sin(2x)dx?

To find the integral of sin(2x)dx, you can use the substitution method. Let u = 2x, then du = 2dx. Rewrite the integral as (1/2)sin(u)du. Using the power rule for integration, this becomes (1/2)(cos(u))/(1) + C = (1/2)cos(2x) + C.

4. What is the derivative of (-1/2)cos(2x) + C?

The derivative of (-1/2)cos(2x) + C is sin(2x), as the derivative of cos(2x) is -sin(2x) and the derivative of C is 0.

5. How can the integral of sin(2x)dx be applied in real-life situations?

The integral of sin(2x)dx can be applied in real-life situations such as calculating the displacement of a simple harmonic motion, finding the average value of a periodic function, and solving problems in physics and engineering involving waves and oscillations.

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