- #1
johann1301
- 217
- 1
Homework Statement
∫cos2x dx
The Attempt at a Solution
I know the answer, and i know how to get there using:
cos2x+sin2x=1
cos2x-sin2x=cos2x
cos2x=(1+cos2x)/2
But why can't i use the chain rule? Can i?
HallsofIvy said:Why would you expect to be able to use it to integrate?
johann1301 said:u=cosx
therefore...
∫(cos2x)dx = ∫(u2)dx = (1/3)u3/u' + C = (1/3)cosx3/(cosx)' + C = (1/3)cosx3/(-sinx) + C
chain rule reversed?
The integral of (cos x)^2 is equal to (1/2)x + (1/4)sin(2x) + C, where C is the constant of integration.
To solve for the integral of (cos x)^2, you can use the trigonometric identity cos^2(x) = (1/2)(1 + cos(2x)). This will allow you to rewrite the integral as a sum or difference of simpler integrals that can be easily solved using integration techniques.
Yes, the integral of (cos x)^2 can be further simplified by using the double angle formula for sine, which is sin(2x) = 2sin(x)cos(x). This will result in the final answer of (1/2)x + (1/4)sin(x)cos(x) + C.
The domain of the integral of (cos x)^2 is all real numbers, as there are no restrictions on the values of x for which the integral can be evaluated.
Yes, the integral of (cos x)^2 can be used to find the area under the curve of the function y = (cos x)^2. This is because the integral represents the accumulation of infinitesimally small rectangles under the curve, which can be used to approximate the total area.