For what values of a is each integral improper? ∫a51x+2dx

Question

For what values of a is each integral improper?   ∫a51x+2dx

Jim Hunt · Accepted Answer

Consider the integral ∫a51x+2dx.  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=−∞  Thus, the given integral is improper for a≤−2 or a=−∞.

scomparve5j · Accepted Answer

∫a51x+2dx=[ln|x+2|]a5  =[ln⁡7−ln|a+2|]  1.946=ln|a+2|  Now is  e1946=a+2  7=a+2  a=7-2  a=5

Vasquez · Accepted Answer

Recall that an integral is improper if one of the limits is $\pm \infty$ 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.
$\frac{1}{x + 2} \Rightarrow x + 2 = 0 \Rightarrow x = - 2$
We determined that the integral is improper for a =-2.

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

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