integrate sin^2xcos^4x dx

Question

getting different answers on every website! please help

Answer 1

∫sin²(x)cos⁴(x)dx

Remember that the half angle identities are:

sin(θ/2) = √(1-cos(θ))/2 so squaring both sides gives:

sin²(θ/2) = (1 - cos(θ))/2 and letting x = θ/2 meaning that θ = 2x we have:

sin²(x) = (1 - cos(2x))/2 = 1/2(1 - cos(2x))

The same holds true for the cosine half angle formula so:

cos²(x) = 1/2(1 + cos(2x))

Substituting these into the integral, we get:

∫[1/2(1 - cos(2x))][1/2(1 + cos(2x))][1/2(1 + cos(2x))]dx

1/8∫[1 - cos(2x)][1 + cos(2x)][1 + cos(2x)]dx

1/8∫[1 - cos²(2x)][1 + cos(2x)]dx

1/8∫[1 - cos²(2x) + cos(2x) - cos³(2x)]dx

1/8∫1dx - 1/8∫cos²(2x)dx + 1/8∫cos(2x)dx - 1/8∫cos³(2x)dx

Let's break this into pieces so we can keep track of it better.

Let this become A - B + C - D where

A = 1/8∫1dx

B = 1/8∫cos²(2x)dx

C = 1/8∫cos(2x)dx

D = 1/8∫cos³(2x)dx

A is trivial to resolve. A = 1/8x

C is pretty easy: C = 1/8[1/2sin(2x)] = 1/16sin(2x)

For B, let's again use the half angle equation we used before

B = 1/8∫cos²(2x)dx

B = 1/8∫1/2(1 + cos(4x))dx

B = 1/16[∫1dx + ∫cos(4x)dx]

B = 1/16[x + 1/4sin(4x)]

B = 1/16x + 1/64sin(4x)

For D, we can use the Pythagorean Identity (sin²(x) + cos²(x) = 1)

D = 1/8∫cos³(2x)dx

D = 1/8∫cos²(2x)cos(2x)dx

D = 1/8∫(1 - sin²(2x))cos(2x)dx

D = 1/8∫cos(2x)dx - 1/8∫sin²(2x)cos(2x)dx

D = 1/16sin(2x) - 1/8∫sin²(2x)cos(2x)dx

To do the right side of this, use a "u" substitution

Let u = sin(2x) then u' = 2cos(2x)

so 1/8∫sin²(2x)cos(2x)dx = (1/8)1/2∫sin²(2x)2cos(2x)dx = 1/16∫u²du = 1/16(1/3u³) = 1/48sin³(2x)

so D = 1/16sin(2x) - 1/48sin³(2x)

Combining:

A - B + C - D

1/8x - (1/16x + 1/64sin(4x)) + 1/16sin(2x) - (1/16sin(2x) - 1/48sin³(2x))

1/8x - 1/16x - 1/64sin(4x) + 1/16sin(2x) - 1/16sin(2x) + 1/48sin³(2x)

1/16x - 1/64sin(4x) + 1/48sin³(2x) + C

By the way, there can be lots of versions of this that are correct. The easiest way to find out if an answer is correct is to plug in integral bounds and see if you get the same numerical answers.

Answer 2

cos^3(2x)/3 + cos(2x)/2 + C

Answer 3

There are multiple ways to do this one. This is the method I thought was the most straightforward.