How do you evaluate the integral of #int (sin^4x)dx#?

Answer 1

Definitively the fastest way to do that, is to use euler formula

#sin^4(x) = (1/(2i)(e^(ix)-e^(-ix)))^4#
#(1/(2i))^4 = (1/2^4)# because #i^4 = 1#

expand with binomial theorem

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

so

#int sin^4(x) dx = 1/2^4int2cos(4x)-8cos(2x)+6dx#

which is

#1/2^4[1/2sin(4x)-4sin(2x)+6x]#
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Answer 2

To evaluate the integral of ( \int \sin^4(x) , dx ), you can use the reduction formula for powers of sine.

First, rewrite ( \sin^4(x) ) as ( (\sin^2(x))^2 ). Then, apply the half-angle identity ( \sin^2(x) = \frac{1 - \cos(2x)}{2} ) to obtain ( \left(\frac{1 - \cos(2x)}{2}\right)^2 ).

Expand this expression and integrate each term separately. After integration, substitute ( u = \cos(2x) ) and proceed with the calculation. Finally, revert to the original variable ( x ) to obtain the final result.

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Answer 3

To evaluate the integral of ∫sin^4(x)dx, you can use the double angle identity for sine, which states that sin^2(x) = (1 - cos(2x))/2. By substituting this identity into sin^4(x), you get (1 - cos(2x))^2/4. Then, expand the squared term and integrate each part separately. The integral simplifies to (3x/8) - (sin(4x)/8) + (sin(2x)/4) + C, where C is the constant of integration.

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Answer 4

To evaluate the integral of (\int \sin^4(x) , dx), you can use the reduction formula or trigonometric identities. One approach is to rewrite (\sin^4(x)) using the power-reducing identity: (\sin^2(x) = \frac{1 - \cos(2x)}{2}). Then, use a double angle formula to further simplify and integrate. Alternatively, you can use trigonometric identities to express (\sin^4(x)) in terms of (\cos(2x)), and then integrate. Both methods will lead to the same result.

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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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