How to Integrate ∫sin^3(x)cos^3(x)dx - odd powers

Integrals of the product of the powers of sine and cosine come in 4 permutations:

1. The powers m and n are both even
2. The powers m and n are even and odd respectively
3. The powers m and n are odd and even respectively
4. The powers m and n are both odd

In this video, we explore case 4 where both powers are odd. The example we use to demonstrate the methodology in this case is...

∫sin³(x)cos³(x)dx

Remember that cos(x) is the first derivative of sin(x), so we reserve one and write the integrand as...

sin³(x)cos³(x) = sin³(x)cos²(x)cos(x)

Then if we write the cos²(x) term as...

cos²(x) = 1 - sin²(x)

we have...

sin³(x)[1 - sin²(x)]cos(x) = [sin³(x) - sin⁵(x)]cos(x)

Then using the substitution u = sin(x), we can easily evaluate the integral.

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