Is something amiss with this integral?

• Jun 2nd 2009, 12:27 PM
Chizum
Is something amiss with this integral?
I just got back from a Calc II test.

There was one question which seemed bizarre; I copied it down and searched the textbook but couldn't find a match.

It's the integral from [0,2] of x^2 + x^-1 + 3^x. (We solve by the Fundamental Theorem of Calculus).

Am I correct in saying that the integral of x^-1 is ln|x|?

If I am, then the resulting evaluation for 0 gives me an undefined answer, correct?
• Jun 2nd 2009, 12:44 PM
Amer
$\displaystyle \int_{0}^{2}x^2+x^{-1}+3^{x} dx$

$\displaystyle \int_{0}^{2}x^2+x^{-1}+3^{x} dx = \frac{x^3}{3} + lnx + \frac{3^x}{ln3}$

it is correct that the integrate of x^-1 is ln(x) take absolute value of x

the other thing is the integrate of 3^x

the integrate of e^x = e^x ln(e) but since ln(e)= 1 so we didn't mention it

that's all
• Jun 2nd 2009, 01:08 PM
Plato
The integral $\displaystyle \int_0^2 {\left( {x^2 + x^{ - 1} + 3^x } \right)dx}$ is an improper integral because $\displaystyle x^{-1}$ is not defined at $\displaystyle x=0$.

Have you studied improper integrals?
• Jun 2nd 2009, 01:17 PM
Chizum
Quote:

Originally Posted by Plato
Have you studied improper integrals?

No, which makes me think my professor must have made a mistake.
• Jun 2nd 2009, 01:38 PM
Plato
Quote:

Originally Posted by Chizum
No, which makes me think my professor must have made a mistake.

Are you sure it was not $\displaystyle \int_{\color{red}1}^2 {\left( {x^2 + x^{ - 1} + 3^x } \right)dx}$ or maybe $\displaystyle x^{\color{red}{-2}}$?
• Jun 2nd 2009, 01:43 PM
Spec
Improper integrals were introduced in Calc I for me. I can't imagine there being a Calc II course were they don't teach you improper integrals, or at least have it as a prerequisite.
• Jun 2nd 2009, 02:04 PM
Chizum
Quote:

Originally Posted by Plato
Are you sure it was not $\displaystyle \int_{\color{red}1}^2 {\left( {x^2 + x^{ - 1} + 3^x } \right)dx}$ or maybe $\displaystyle x^{\color{red}{-2}}$?

Pretty sure, I spent at least 5 minutes thinking about it and then copied it down onto a piece of scratch paper.

@ Spec,

The text we're using doesn't seem to introduce Improper Integrals till 8.8 (according to the index). We're on like 6.5 right now.