Sine and cosine: odd and even powers

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

If a power is odd, reserve one factor and convert the others using sin²+cos²=1. If both powers are even, use power-reduction identities.

Step-by-step example

1. Reserve sin x and convert sin²x.

   $$\int\sin^3x\cos^2x\,dx=\int(1-\cos^2x)\cos^2x\sin x\,dx$$

2. Use u = cos x with a minus sign.

   $$I=-\int(u^2-u^4)\,du$$

3. Integrate and return to cos x.

   $$I=-\frac{\cos^3x}{3}+\frac{\cos^5x}{5}+C$$