Question

I think I may have the answer but want some feedback: Integrate 73arctan(1/x) dx. I used integration by parts first, then u-substitution for the 2nd integral. Can someone work out the same problem so I can check my work? Thanks!

Answer 1

\[73x*\arctan(1/x)+1/2\ln \left| x^2+1 \right|+C\]

Answer 2

Oh and it's a definite integral\[\int\limits\limits_{1}^{\sqrt{3}}73\tan^{-1} (1/x)\]

Answer 3

Great, I got that before doing the definite integral part... could you please show me this also?

Answer 4

\[73\pi \sqrt{3}/6 - 73\pi /4 + 73/2 * \ln 2\] is as far as I went... is this right?

Answer 5

whats your final answer

Answer 6

What I just wrote above... my instructor doesn't ask us to multiply out pi or natural log unless it's a simple answer like 1 or 0.

Answer 7

let me work it out hold on 1 sec

Answer 8

something might be off

Answer 9

Oh by the way, I got 73/2 before the ln(x^2+1)

Answer 10

let me try and draw out my steps for you okay, so that we can compare

Answer 11

ok sounds good

Answer 12

hold on i left out the constat

Answer 13

Ok, looks like we have the same u and dv when i saw the post for a second. Thanks for taking the time to help me, the constant & the sqrt(3) were kind of throwing me off :/ Done the problem like 5 times. If it wasn't a definite integral it wouldn't be as confusing.

Answer 14

let u=arctan(1/x) and dv=dx

so v=x and du=dx(-1/x^2)/(1+(1/x^2))= -dx/(x^2+1)

so INT[arctan(1/x)dx] = x arctan(1/x) - INT[-xdx/(x^2+1)]

the integral on the right is of the form dz/z where z=x^2+1

so INT[arctan(1/x)dx]= x arctan(1/x) + (1/2)ln|x^2+1| +C

evaluating the integral and right side of the equation from
1 to SQRT(3) you get

73(SQRT(3)arctan(1/SQRT(3)) +(1/2)ln4) - 73(arctan(1) - (1/2)ln2)

73((PI)/6)SQRT(3) + ln2 - (PI)/2 - (1/2)ln2)

Answer 15

So you basically factored out the 73?

Answer 16

yeah, cause i mean its a constant you know, you can plug it back in. But i think we have the same answers yeah

Answer 17

Would it be safe to say when I encounter a similar integral, that I can leave out the constant til the end and it will be k(x*arctan (1/x) + 1/2ln(x^2+1)) +C?

Answer 18

you can leave out the constant when integrating, but when you evaluate make sure that the constant dosent affect your final answer. But since your teacher asked you not to fully simplify i would say it is safe to leave it out and then just factor it back in

Answer 19

Ok great, that is going on my list of integrals-I memorize-because-I-never-want-to-integrate-it-again. :) Thank you!