How do you find the integral of #int (cosx)^4 dx#?

Answer 1

The answer is #=1/32sin(4x)+1/4sin(2x)+3/8x+C#

First, linearize #cos^4x# by applying Euler's Identity
#cosx=(e^(ix)+e^(-ix))/2#

Consequently,

#cos^4x=((e^(ix)+e^(-ix))/2)^4#
#=1/16(e^(4ix)+e^(-4ix)+4e^(2ix)+4e^(-2ix)+6)#
#=1/8(cos(4x)+4cos(2x)+3)#

Consequently,

#intcos^4xdx=1/8intcos(4x)dx+1/2intcos(2x)+3/8int1dx#
#=1/32sin(4x)+1/4sin(2x)+3/8x+C#
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Answer 2

To find the integral of ( \int (\cos x)^4 , dx ), you can use trigonometric identities and integration techniques. One common approach is to use the reduction formula for powers of cosine.

  1. Start by applying the power-reducing identity for cosine: [ (\cos x)^4 = (\cos^2 x)^2 = (\frac{1 + \cos(2x)}{2})^2 ]

  2. Expand ( (\frac{1 + \cos(2x)}{2})^2 ) and simplify.

  3. Integrate the resulting expression.

  4. Finally, don't forget to include the constant of integration ( + C ).

If you need further assistance with the specific steps or if you have any questions regarding the process, feel free to ask!

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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.

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.

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.

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.

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