Improper Integral

#Calc-2#Math#Integrals Improper Integrals let us find the area under an entire curve, or from a point to infinity.

${\int }_a^{\infty }\frac{1}{x}dx$ These integrals are solved by replacing infinity with a $b$ or some other variable and then doing ${\lim }_{b\to \infty }{\int }_a^{b}f(x)dx$ Improper Integrals are Convergent if they result in a number, they are Divergent if they result in plus or minus infinity.

Improper integrals where $f(x)=\frac{1}{x^p}$ will converge if $p>1$ and will diverge if $p\le 1$

Another kind of improper integral is when the function becomes undefined along the integral bounds. e.g. ${\int }_{-2}^3\frac{1}{x^4}dx$ In this case, split up the integral and do the limit as a variable approaches the spot where the function becomes undefined. For the above example, ${\lim }_{b\to 0^{-}}{\int }_{-2}^b\frac{1}{x^4}dx+{\lim }_{a\to 0^{+}}{\int }_a^3\frac{1}{x^4}dx$ Solve both by finding the limit

Sometimes we can do a Comparison Test to quickly determine if an improper integral converges. They function the same as direct comparison tests for series, take a look at that page, except we compare improper integrals instead of series’.