Trig sub is used when you see a square root involving a quadratic expression that matches a Pythagorean identity
Main Ideas
- Square roots with quadratics replaced with trig identities
- Integrating in θ and then converting back to x using triangles or SOH CAH TOA
| Expression |
Substitution |
Derivative |
| a^2-x^2 |
x=asinθ |
dx=acosθdθ |
| a^2+x^2 |
x=atanθ |
dx=asec^2θdθ |
| x^2-a^2 |
x=asecθ |
dx=asecθtanθdθ |
How To:
- Identify the pattern
- Is it $a^2-x^2$ or $a^2+x^2$ or even $x^2-a^2$ ?
- Rewrite dx
- Once you know the pattern and know x, find dx
- Rewrite the integral in terms of θ
- Simplify any trig expressions
- Use known identities if you see something like $sin^2x$ or a double angle
- Integrate with respect to θ
- Convert back to original variable