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

Question

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

Luke Alvarez · Accepted Answer

# int cos^4xdx = (sinxcos^3x)/4 + 3/8(cosxsinx) + 3/8x#

Write: #cos^4x = cos^3x*cosx# and integrate by parts:

#int cos^4xdx = int cos^3x cosx dx = int cos^3x d(sinx)#

#int cos^4xdx = sinxcos^3x - int sinx d(cos^3x)#

#int cos^4xdx = sinxcos^3x + 3int sin^2x cos^2xdx#

Now use the identity:

#sin^2x = 1-cos^2x#

#int cos^4xdx = sinxcos^3x + 3int (1-cos^2x) cos^2xdx#

#int cos^4xdx = sinxcos^3x + 3int cos^2x dx -3int cos^4xdx#

We have now the same integral on both sides and we can solve for it:

#4 int cos^4xdx = sinxcos^3x + 3int cos^2x dx #

# int cos^4xdx = (sinxcos^3x)/4 + 3/4int cos^2x dx #

Using the same process:

#int cos^2x dx = int cosxd(sinx) = cosxsinx + int sin^2xdx#

#int cos^2x dx = int cosxd(sinx) = cosxsinx + int (1-cos^2x)dx#

#int cos^2x dx = int cosxd(sinx) = cosxsinx + x - int cos^2xdx#

#int cos^2x dx = (cosxsinx)/2 + x/2#

Substituting in the expression above:

# int cos^4xdx = (sinxcos^3x)/4 + 3/8(cosxsinx) + 3/8x#

Evelyn Agard · Answer

undefined To find the integral of cos^4(x) dx, you can use the reduction formula for even powers of cosine:

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

Applying this formula to cos^4(x), you get:

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

Now, for ∫cos^2(x) dx, you can use the identity cos^2(x) = (1 + cos(2x))/2:

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

Substitute this result back into the integral of cos^4(x) to get the final answer:

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