How do you find the antiderivative of # cos^4 (x) dx#?

Answer 1

You want to split it up using trig identities to get nice, easy integrals.

#cos^4(x) = cos^2(x)*cos^2(x)#
We can deal with the #cos^2(x)# easily enough by rearranging the double angle cosine formula.
#cos^4(x) = 1/2(1 + cos(2x))*1/2(1 + cos(2x))#
#cos^4(x) = 1/4(1 + 2cos(2x) + cos^2(2x))#
#cos^4(x) = 1/4(1 + 2cos(2x) + 1/2(1 + cos(4x)))#
#cos^4(x) = 3/8 + 1/2*cos(2x) + 1/8*cos(4x)#

So,

#int cos^4(x) dx = 3/8*int dx + 1/2*int cos(2x) dx + 1/8*int cos(4x) dx#
#int cos^4(x) dx= 3/8x + 1/4*sin(2x) + 1/32*sin(4x) + C#
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Answer 2

To find the antiderivative of cos^4(x) dx, you can use the reduction formula for powers of cosine. The reduction formula states that:

∫cos^n(x) dx = (cos^(n-1)(x) * sin(x))/n + (n-1)/n * ∫cos^(n-2)(x) dx

Using this formula, you can find the antiderivative of cos^4(x) dx. Start by setting n = 4:

∫cos^4(x) dx = (cos^3(x) * sin(x))/4 + (3/4) * ∫cos^2(x) dx

Now, you need to simplify ∫cos^2(x) dx using the reduction formula again. Set n = 2:

∫cos^2(x) dx = (cos(x) * sin(x))/2 + (1/2) * ∫dx ∫cos^2(x) dx = (cos(x) * sin(x))/2 + (1/2) * x + C

Now, substitute this result back into the first equation:

∫cos^4(x) dx = (cos^3(x) * sin(x))/4 + (3/4) * ((cos(x) * sin(x))/2 + (1/2) * x) + C ∫cos^4(x) dx = (cos^3(x) * sin(x))/4 + (3/8) * cos(x) * sin(x) + (3/8) * x + C

So, the antiderivative of cos^4(x) dx is (cos^3(x) * sin(x))/4 + (3/8) * cos(x) * sin(x) + (3/8) * x + 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.

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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