crealolobk

2021-12-31

For what values of a is each integral improper?
${\int }_{a}^{5}\frac{1}{x+2}dx$

Jim Hunt

Consider the integral ${\int }_{a}^{5}\frac{1}{x+2}dx$.
Check whether the provided function is improper or not, find the point of infinite discontinuity.
Equate the denominator to zero.
x+2=0
x=-2
The function has a discontinuity at x=-2, so the integral is improper on the interval [a,5] for C.
And the integral is improper when $a=-\mathrm{\infty }$
Thus, the given integral is improper for $a\le -2$ or $a=-\mathrm{\infty }$.

scomparve5j

${\int }_{a}^{5}\frac{1}{x+2}dx={\left[\mathrm{ln}|x+2|\right]}_{a}^{5}$
$=\left[\mathrm{ln}7-\mathrm{ln}|a+2|\right]$
$1.946=\mathrm{ln}|a+2|$
Now is
${e}^{1946}=a+2$
7=a+2
a=7-2
a=5

Vasquez

Recall that an integral is improper if one of the limits is $±\mathrm{\infty }$ or it has infinite discontinuities in the interval of integration. In this example, value a is a value at which graph of $y=\frac{1}{x+2}$ is discontinuous. We can easily determine that graphs of rational functions have a discontinuity when the denominator is equal to 0.
$\frac{1}{x+2}⇒x+2=0⇒x=-2$
We determined that the integral is improper for a =-2.

Do you have a similar question?