Definition: An integral where the lower and/or upper bound is infinite are called improper integrals.
Some examples of improper integrals: $\int_0^\infty \frac{1}{x}dx$ and $\int_{-\infty}^1 e^{2x}dx$.
Step plan: In order to calculate an improper integral you follow the following three steps.
- Replace the improper integral by a variable integration bound;
- Calculate this integral;
- Consider what happens when the variable integration bound goes to infinity.
Example
Consider the improper integral $\int_{-\infty}^1 e^{2x}dx$.
- The use of the variable integration bound $t$ results in $\int_t^1 e^{2x}dx$;
- $\int_t^1 e^{2x}dx=[\frac{1}{2}e^{2x}]_{x=t}^{x=1}=\frac{1}{2}e^2-\frac{1}{2}e^{2t}$;
- If $t\rightarrow -\infty$, then $\frac{1}{2}e^2-\frac{1}{2}e^{2t} \rightarrow \frac{1}{2}e^2-0=\frac{1}{2}e^2$.
Conclusion: $\int_{-\infty}^1 e^{2x}dx=\frac{1}{2}e^2$.