# How do I evaluate #int cos^5(x) sin^4(x) dx#?

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To evaluate the integral (\int \cos^5(x) \sin^4(x) , dx), you can use trigonometric identities and integration by substitution.

First, rewrite (\sin^4(x)) using the identity (\sin^2(x) = 1 - \cos^2(x)). This gives: (\sin^4(x) = (\sin^2(x))^2 = (1 - \cos^2(x))^2).

Then, use the substitution (u = \cos(x)), so (du = -\sin(x) , dx).

This transforms the integral into: (\int u^5 (1 - u^2)^2 , (-du)).

Expand and integrate this expression. Finally, resubstitute (u = \cos(x)) to find the final result.

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To evaluate ( \int \cos^5(x) \sin^4(x) , dx ), you can use trigonometric identities to simplify the expression. One common approach is to use the power-reducing identities, which express higher powers of sine and cosine in terms of lower powers. After simplifying, you'll end up with a polynomial expression that can be integrated using standard techniques.

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