Integrating sin² and cos²
There is no standard rule that directly integrates or . First rewrite the square using a double-angle identity; then integrate the resulting constant and cosine terms.
Convert the square before integrating
Build a bridge to a known rule
not ready
sin²(kx)
↓ identity first
½[1 − cos(2kx)]
the angle doubles
↓
integrate term by term
For cos²(kx), use a plus. Divide the new cosine by 2k.
Here stands for the whole original angle. For example, if the expression is , then and . The sign distinguishes the pair: sine squared uses minus; cosine squared uses plus. The angle inside the cosine doubles.
Track the outside half and the doubled angle
Worked example
Integrate a squared function with angle 2x
The factor combines the outside with division by the new inside coefficient 4.
Common mistake
Rewrite tangent squared as secant squared minus one
The identity can be rearranged to . This creates two terms with direct antiderivatives: tangent squared becomes secant squared minus one.
Worked example
Integrate tangent squared
Common mistake
Use the identity only when the square blocks the rule
- already has a standard rule; do not rewrite it.
- needs the squared identity first.
- Keep exact values until the requested final approximation.
Evaluate exactly.