Int(sin^3 x cos^3 x)dx?

Question

integral of sin^3 x times cos^3 x

Answer 1

∫(sinx)^3 (cosx)^3 dx

We will be using trigonometric identities here and now to solve this integral.

∫(sin x)^3 (cos x)^3 dx

= ∫(sin x)^3 (cos x)^2 (cos x) dx

= ∫(sin x)^3 (1 - (sin x)^2) (cos x) dx, because (sin x)^2 + (cos x)^2 = 1

We use a substitution. Let u = sin x, and du = cos x dx.

= ∫(sin x)^3 (1 - (sin x)^2) (cos x) dx

= ∫u^3 (1 - u^2) du

= ∫(u^3 - u^5) du

= u^4 / 4 + u^6 / 6 + C

= (sin x)^4 / 4 + (sin x)^6 / 6 + C

This is your solution.

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