AP Calculus BC / Unit 6: Integration and Accumulation of Change
6.10 Integrating Functions Using Long Division and Completing the Square
Notes
- If the top's degree is at least the bottom's, divide first: $\displaystyle \frac{x^2+1}{x}=x+\frac{1}{x}$.
- Complete the square to get the arctan form: $\displaystyle \int\frac{dx}{(x-h)^2+a^2}=\frac{1}{a}\arctan\frac{x-h}{a}+C$.
Practice
$\displaystyle \int\frac{x^2+1}{x}\,dx=$
Divide: $\displaystyle x+\frac{1}{x}$.
Completing the square on $x^2+4x+5$ gives:
$x^2+4x+4+1$.
$\displaystyle \int\frac{dx}{x^2+4x+5}=$
$\displaystyle \int\frac{du}{u^2+1}$ with $u=x+2$.
Ready for more? Take the Unit 6 test.