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

\int \sin^3x\cos^3x\mathrm{d}x

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

\sin^3x\cos^3x=\sin^3x\cos^2x\cos x

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

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

we have...

\sin^3x\left [ 1-\sin^2x \right ]\cos x=\left [ \sin^3x-\sin^5x \right ]\cos x

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

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