# How do you find the integral of #int sin^5(x)cos^8(x) dx#?

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To find the integral of (\int \sin^5(x)\cos^8(x) , dx), you can use trigonometric identities and integration by parts. Here's how:

- Start by using the identity (\sin^2(x) = 1 - \cos^2(x)) to rewrite (\sin^5(x)) as ((1 - \cos^2(x))^2\sin(x)).
- Expand ((1 - \cos^2(x))^2) using the binomial theorem.
- Now you have an integral involving powers of (\sin(x)) and (\cos(x)), which you can integrate using integration by parts.
- Let (u = \cos(x)) and (dv = \sin^4(x)\cos^4(x) , dx), then differentiate (u) to find (du) and integrate (dv) to find (v).
- Apply integration by parts formula: (\int u , dv = uv - \int v , du).
- Substitute the values of (u), (du), (dv), and (v) into the integration by parts formula and evaluate the integral.
- Repeat the process if necessary until you obtain an expression that can be easily integrated.

Following these steps will lead you to the solution of the integral (\int \sin^5(x)\cos^8(x) , dx).

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