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

TR Smith is a product designer and former teacher who uses math in her work every day.

Joined: 5 years agoFollowers: 215Articles: 455
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

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The 4th powers have been replaced by 2nd powers, leaving us with easier functions to integrate.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

2

9

2

4

4

12

10

29