Calculating the integral: ∫7sin^4(x)cos^2(x)dx

Answer 1

You can use the Pythagorean identity sin²(x) + cos²(x = 1 to get cos²(x) = 1 - sin²(x) and then substitute to yield:

∫7sin⁴(x)(1 - sin²(x)) dx

7∫sin⁴(x) dx - 7∫sin⁶(x) dx

Then you can use the power reduction identities (which are variant of the double angle identities) which go like this:

You can square both sides of the "sine" identity (getting an identity for sin⁴(x)) and you can cube both sides as well (getting an identity for sin⁶(x)).

You can also use a form of the triple angle identity:

cos³(2x) = (1/4)cos(6x) + (3/4)cos(3x)

All these will allow you to simplify the integral into a series of simple cosine integrals.

Answer 2

This graph also shows tangent lines to the graph of the antiderivative.

desmos.com/calculator/oarbfxy7ma

Answer 3

Hi Lily!

This looks like an indefinite integral - there are no lower or upper bounds given. I recommend getting started by typing this into Wolfram Alpha. The first step converts cosine squared into (one minus sine square). The next step is likely distribution and then splitting the integral.

Best,

Will A.