Question

Please help me find the anti-derivative of x/(x^2+1)

Answer 1

I'm using the shell method to find the volume of the solid generated when the region enclosed by
\[y=\frac{ 1 }{ x^2+1 },y=0 \]
\[x=0,x=1\]
is revolved about the y-axis

Answer 2

this is where i am in the problem, just need some help integrating
\[2\pi \int\limits_{0}^{1}\frac{ x }{ x^2+1 }dx\]

Answer 3

Let's write it this way, maybe it'll be easier :)
\[\pi \int\limits\limits_{0}^{1}\frac{ 2x }{ x^2+1 }dx\]
better?

Answer 4

hmm...still stuck

Answer 5

lol, if you take that denominator, its derivative is to be found somewhere in the integrand...

Answer 6

ok, the derivative of the denominator is 2x

Answer 7

Which is the numerator... can you see it now? :)

Answer 8

ln(x^2+1)

Answer 9

That's good, now evaluate it from x = 0 to x = 1
And then multiply it by pi, and you'll have your result :)

Answer 10

So multiplying the integrand by two before integrating doesn't effect the final result negatively?

Answer 11

I didn't multiply the integrand by 2, at least, I didn't do JUST that, remember, the number outside the integrand was 2pi?
I just brought the 2 inside the integrand (you can do that with constants) so you could see that the derivative of the denominator can be found in the integrand.

Answer 12

ok thanks