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How do you find the integral of #cos^3 2x dx#?

Answer 1

Ezra Adams

# int cos^3 2xdx=1/24(sin6x+9sin2x)+C.#

Recall that, #cos3theta=4cos^3theta-3costheta.#

#:. cos3theta+3costheta=4cos^3theta, or, #

# cos^3 theta=1/4(cos3theta+3costheta).#

Replacing #theta# by #2x,# we have,

#cos^3 2x=1/4(cos 6x+3cos 2x).#

#:. int cos^3 2xdx=int{1/4(cos 6x+3cos 2x)}dx,#

#=1/4intcos 6x dx+3/4intcos2xdx,#

#=1/4*sin(6x)/6+3/4*sin(2x)/2,#

#=1/24*sin6x+3/8*sin2x,#

# rArr int cos^3 2xdx=1/24(sin6x+9sin2x)+C.#

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Answer 2

Luke Alvarez

see below

Another approach

#intcos^3 2xdx#

#=intcos2xcos^2 2xdx#

#=intcos2x(1-sin^2 2x)dx#

#=int(cos2x-sin^2 2xcos2x)dx#

integrating by inspection

#= 1/2sin2x-1/6sin^3 2x+C#

which can be shown by trig identities to be equivalent to the previous solution. This is left for the reader to verify

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Answer 3

Cameron Alexander

To find the integral of cos^3(2x) dx, you can use the trigonometric identity cos^3(x) = (1/4)(3cos(x) + cos(3x)). Apply this identity to cos^3(2x), integrate, and then substitute back for 2x. The result will be:

∫cos^3(2x) dx = (1/8)sin(2x) + (3/32)sin(6x) + C

Where C is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.