AP Calculus BC / Unit 6: Integration and Accumulation of Change
6.13 Evaluating Improper Integrals
Notes
An improper integral has an infinite limit or an integrand that blows up. Write it as a limit:
$$\int_1^\infty f(x)\,dx=\lim_{b\to\infty}\int_1^b f(x)\,dx$$
- If the limit is finite, it converges; otherwise it diverges.
- $\displaystyle \int_1^\infty\frac{1}{x^p}dx$ converges when $p>1$.
Practice
$\displaystyle \int_1^\infty\frac{1}{x^2}dx=$
$\displaystyle \lim\left(1-\frac{1}{b}\right)=1$.
$\displaystyle \int_1^\infty\frac{1}{x}dx$:
$\ln b\to\infty$.
$\displaystyle \int_0^1\frac{1}{\sqrt x}dx=$
$\displaystyle 2\sqrt x$ from 0 to 1.
Ready for more? Take the Unit 6 test.