Question

evaluate the improper integral, state if it diverges:

integral (0 -> infinity) (2dx/25+x^2)

Answer 1

Ah I know the trick for this one, but first for your understanding: Tell me what the derivative of x^2+25 is, cool? :-)

Answer 2

2x-25x

Answer 3

Erm nope :-)

How about what the derivative of arctangent is? That's also known as tan\(^{-1}\)( )

Answer 4

It's just 2x dx for the first one ;-)

Answer 5

youre right, sorry i confused that with integration, ok so its 2x dx

Answer 6

arctan is 1/(1+x^2)

Answer 7

I think what I'm trying to get you to see is two things here... (goes to write) And yes that's correct! gj

Answer 8

\[\int\limits \frac{2 dx}{25+x^2} = 2 \int\limits \frac{1}{x^2+5^2} dx\]

Answer 9

scoot that junk out of the numerator so you can see it better

Answer 10

yeah i see that looks much more beautiful. so its arctan.

Answer 11

And now for the magic bullet!

\[\frac{d}{dx} \frac{1}{x^2+a^2}=\frac{1}{a}\tan^{-1}( \frac{x}{a}) dx\]

Answer 12

Careful to not forget the 2 outside the integrand though for your final answer. Got it now? :D

Answer 13

yeah i do so i now have to look at the end behavior of ArcTan as it goes to Infinity

Answer 14

And that's easy to do if you remember this: inverse switches domain & range

So what's the domain for tangent?

Answer 15

-pi/2 to pi/2

Answer 16

Yep, guess which is \(\infty\) and which is -\(\infty\)? ;-)

Answer 17

For arctan

Answer 18

well the range of tan would be the domain of arc tan correct?

Answer 19

Yeah the range of tangent is all reall #'s :-)

Answer 20

The domain of tangent however is restricted

Answer 21

Remember inverse function are the same as the original in shape except they have been flipped over the line y=x and that means their domains and ranges are swapped compared to the original

\(\infty \leftarrow\rightarrow\frac{\pi}{2}\)