Integrating with Trig Identities

Starting point: what you already know

• Double angle formulas (Compound Angles and Double Angles): $\cos 2x=1-2{\sin }^{2}x=2{\cos }^{2}x-1$ and $\sin 2x=2\sin x\cos x$.
• Pythagorean identities (Reciprocal Trigonometric Functions): ${\sin }^{2}x+{\cos }^{2}x=1$ and $1+{\tan }^{2}x={\sec }^{2}x$.
• Integrals of trig functions from Mathematics Advanced: $\int \cos axdx=\frac{1}{a}\sin ax+C$, $\int \sin axdx=-\frac{1}{a}\cos ax+C$, $\int {\sec }^{2}axdx=\frac{1}{a}\tan ax+C$.
• Radians: calculus with trig always uses radians.

Rewrite the squares

Rearrange the double angle formulas for $\cos 2x$:

Now each one is easy to integrate:

$$\int {\sin }^{2}xdx=\frac{x}{2}-\frac{\sin 2x}{4}+C\int {\cos }^{2}xdx=\frac{x}{2}+\frac{\sin 2x}{4}+C$$

Other products and powers

Quick summary

• ${\sin }^{2}x=\frac{1}{2}(1-\cos 2x)$ and ${\cos }^{2}x=\frac{1}{2}(1+\cos 2x)$.
• $\int {\sin }^{2}xdx=\frac{x}{2}-\frac{\sin 2x}{4}+C$ and $\int {\cos }^{2}xdx=\frac{x}{2}+\frac{\sin 2x}{4}+C$.
• $\sin x\cos x=\frac{1}{2}\sin 2x$ and ${\tan }^{2}x={\sec }^{2}x-1$.
• For ${\sin }^{2}kx$, the new angle is $2kx$.
• Odd powers: keep one factor and substitute.

Quiz