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How to Integrate sin(x)^4 and cos(x)^4, Fourth Power Trig Functions

Graphs of f(x) = 2 + cos(x)^4 (blue) and g(x) = sin(x)^4 (red).
Graphs of f(x) = 2 + cos(x)^4 (blue) and g(x) = sin(x)^4 (red).

Like other powers of trig functions, the antiderivatives of sin(x)^4 and cos(x)^4 [also written sin^4 (x) and cos^4 (x)] can be found by reducing the trigonometric integral into simpler forms using fundamental trig identities. Solving integrals this way eliminates the need for complicated u-substitution, and allows you to simplify even the most seemingly intractable trigonometric function integrals. Here is a step-by-step tutorial.


Step 1

The first step in integrating sin(x)^4 and cos(x)^4 is to factor them as sin(x)^2*sin(x)^2 and cos(x)^2*cos(x)^2 and then apply the trig identities

sin(x)^2 = 1 - cos(x)^2
cos(x)^2 = 1 - sin(x)^2

This gives you

sin(x)^4 = sin(x)^2*(1 - cos(x)^2)
= sin(x)^2 - sin(x)^2*cos(x)^2

cos(x)^4 = cos(x)^2*(1 - sin(x)^2)
= cos(x)^2 - sin(x)^2*cos(x)^2

The next steps will involve applying more conversion formulas to further simplify the original functions. Here are the key equations used in this tutorial.

Trig identities and formulas
Trig identities and formulas

Step 2

The equivalent forms may seem more complicated since they now have terms of sine multiplied by cosine. However, these mixed terms can be reduced using the double-angle identity

sin(2x) = 2*sin(x)*cos(x)

This implies that

sin(x)^2*cos(x)^2
= [sin(x)*cos(x)]^2
= [0.5*sin(2x)]^2
= 0.25*sin(2x)^2

Now we can simplify our new forms of sin(x)^4 and cos(x)^4:

sin(x)^4
= sin(x)^2 - sin(x)^2*cos(x)^2
= sin(x)^2 - 0.25*sin(2x)^2

cos(x)^4
= cos(x)^2 - sin(x)^2*cos(x)^2
= cos(x)^2 - 0.25*sin(2x)^2

The 4th powers have been replaced by 2nd powers, leaving us with easier functions to integrate.


Step 3

The last step in transforming the function is to apply one more round of double-angle formulas to get rid of the powers of 2. This will leave us with the simplest integral possible. The key double-angle formulas are

cos(x)^2 = 0.5 + 0.5*cos(2x)
sin(x)^2 = 0.5 - 0.5*cos(2x)

Applying these trig identities gives us

sin(x)^4
= sin(x)^2 - 0.25*sin(2x)^2
= [0.5 - 0.5*cos(2x)] - 0.25[0.5 - 0.5*cos(4x)]
= 0.375 - 0.5*cos(2x) + 0.125*cos(4x)

cos(x)^4
= cos(x)^2 - 0.25*sin(2x)^2
= [0.5 + 0.5*cos(2x)] - 0.25[0.5 - 0.5*cos(4x)]
= 0.375 + 0.5*cos(2x) + 0.125*cos(4x)

And now we have finally eliminated all exponents from the original forms of the functions using basic trig identities.


Step 4

At this point, we can find the antiderivatives of sin(x)^4 and cos(x)^4 using the alternative forms with doubled and quadrupled angles.

∫ sin(x)^4 dx
= ∫ [0.375 - 0.5cos(2x) + 0.125cos(4x)] dx
= 0.375x - 0.5sin(2x)/2 + 0.125sin(4x)/4 + c
= 0.375x - 0.25sin(2x) + 0.03125sin(4x) + c
= [12x - 8*sin(2x) + sin(4x)]/32 + c

∫ cos(x)^4 dx
= [0.375 + 0.5cos(2x) + 0.125cos(4x)] dx
= 0.375x + 0.5sin(2x)/2 + 0.125sin(4x)/4 + c
= 0.375x + 0.25sin(2x) + 0.03125sin(4x) + c
= [12x + 8*sin(2x) + sin(4x)]/32 + c

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