Reviewed by Phoebe Belza-Barrientos
How to integrate (sin^2)x?
Answer: The integral of (sin^2)x is (x/2) - (sin(2x)/4) + C
Integrating (sin^2)x involves finding the antiderivative of the square of the sine function. This is a common problem in calculus that can be solved using a trigonometric identity to simplify the process. The solution typically involves expressing (sin^2)x in a different form that is easier to integrate.
Methods
Math Tutor Explanation Using the Power-Reducing Identity
The power-reducing identity rewrites the square of sine in terms of cosine of double the angle, making the integral straightforward.
Step 1: Step 1: Use the identity (sin^2)x = (1 - cos(2x))/2
Step 2: Step 2: Set up the integral as ∫(1 - cos(2x))/2 dx
Math Tutor Explanation Using Integration by Parts
You can use integration by parts by letting u = sin x and v = sin x, though this approach is more involved and is less efficient than using identities.
Step 1: Step 1: Let u = sin x and dv = sin x dx
Step 2: Step 2: Compute du = cos x dx and v = -cos x
Step 1:
Step 2:
Math Tutor suggests: Practice More Trigonometric Integrals
Expand your skills in integrating trigonometric functions with these related problems and solutions.
FAQ on Integrating Trig Squares
What is the power-reducing identity for sin^2x?
It is (sin^2)x = (1 - cos(2x))/2
Can you integrate (sin^2)x directly?
It's simpler to use trigonometric identities to rewrite the function before integrating
What is the integral of (cos^2)x?
The integral is (x/2) + (sin(2x)/4) + C
Why is the constant of integration '+ C' added?
Because indefinite integrals represent a family of functions differing by a constant
Are there other ways to solve the integral of (sin^2)x?
Yes, integration by parts is possible, but using the identity is most efficient
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