AP Calculus BC / Unit 6: Integration and Accumulation of Change
6.14 Selecting Techniques for Antidifferentiation
Notes
Decide by the form:
- Basic rule? Use it. Inside function with its derivative? Substitution.
- Rational with higher top degree? Long division. Quadratic bottom? Complete the square.
- Product of unlike types (BC)? Parts. Factorable bottom (BC)? Partial fractions.
Practice
Best method for $\displaystyle \int\frac{x}{x^2+9}\,dx$?
The top is half the derivative of the bottom.
$\displaystyle \int\frac{x}{x^2+9}\,dx=$
$\displaystyle \frac{1}{2}\int\frac{du}{u}$.
Best method for $\displaystyle \int\frac{1}{x^2+9}\,dx$?
There's no $x$ on top, so it's $\displaystyle \frac{1}{3}\arctan\frac{x}{3}$.
Ready for more? Take the Unit 6 test.