Chapter 49: Improper Integrals

We have dealt with definite integrals like:

25xex2dx\int_{2}^{5}x e^{x^{2}}dx

Where the domain and range are both finite.


Definite Integral - is Improper if- the domain or range is unbounded.


Example

1exdx-\int_{1}^{\infty}e^{-x}dx

=limt1texdx=\lim_{t\rightarrow \infty}-\int_{1}^{t}e^{-x}dx

=limt[ex]1t=\lim_{t\rightarrow \infty}[-e^{-x}]_{1}^{t} 

=limt[ex][e1]]=\lim_{t\rightarrow \infty}[-e^{-x}]-[-e^{-1}]] (goes to 0 as tt \rightarrow \infty)

=e1=e^{-1}

\therefore  The region is unbounded but it has finite total area (and infinite perimeter)


Example

Discuss

Find 0CS11+x2dx\int_{0}^{CS} \frac{1}{1+x^{2}}dx

  • Method- to find CCSf(x)dx\int_{C}^{CS}f\left ( x \right )dx =limtCf(x)dx= lim \int_{t}^{C}f\left ( x \right )dx

tCSt\rightarrow CS

or CSCf(x)dx\int_{-CS}^{C}f\left ( x \right )dx =limCtf(x)dx= lim \int_{C}^{t}f\left ( x \right )dx

tCSt\rightarrow CS


Example

CSCSexe2x+1dx\int_{-CS}^{CS} \frac{e^{x}}{e^{2x+1}}dx

Spit up the integral (reduce to known cases)

0CSexe2x+1dx+CS0exe2x+1dx\int_{0}^{CS} \frac{e^{x}}{e^{2x+1}}dx+\int_{-CS}^{0}\frac{e^{x}}{e^{2x+1}}dx

Apply substitution u=exdu=exu= e^{x}\Rightarrow du=e^{x}

An improper integral is Convergent if,

It is Divergent otherwise,


Example

1CS1xdx\int_{1}^{CS}\frac{1}{x}dx

Thus, the integral is divergent.


Question

When is 1CS1xdx\int_{1}^{CS}\frac{1}{x}dx convergent?

We need p>1p> 1 for this limit to be finite

We get-


Fact: If p>1p> 1 , then 1CS1xdx=1p1\int_{1}^{CS}\frac{1}{x}dx= \frac{1}{p-1}


Unbounded Range

Fact: if f(x)f(x) has a unique asymptote

Thus, the total area is finite.






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