Question

Could someone help me integrate ((dx)/(x^2+1)^2) on the interval 0 to 1?

Answer 1

by parts ?

Answer 2

We are using trigonometric substitution to solve. So far I have \[\int\limits_{0}^{1}\frac{ 1 }{ \sqrt{^{2}+1} }dx\] with \[x=\csc \theta dx=-\csc \theta*\cot \theta\]

Answer 3

I believe it's by part since that is what we just finished doing in the last section, however we are supposed to, and I quote, " transform the problem
into an equivalent problem involving trig functions of an angle theta.
You'll notice that three of the problems are definite integrals (that
is, they have limits of integration). For those problems, you will
also have to transform the limits of integration so that they are
correct in the new "universe" relative to theta."

Answer 4

try \(u=\tan(x)\) to make it easier

Answer 5

would tan(theta)=1/(sqrt(x^2=1))?

Answer 6

x= tan y
x^2+1 = sec^2 y
1/sec y = cos y
so now u only have integral of cos y
and change the limits accordingly.........

Answer 7

Thank you for the help up to this point. It was much easier to use tany rather than cscy. I'm still not clear on how I would go about changing the limits. Any suggestions? Thanks again.

Answer 8

wait,
u got integral of cos x ?
u need to change dx to dy also
x= tan y
dx = sec^2 y dy
so integral of cos y sec^2 y dy
which is integral of sec y dy
can u solve integral of sec y ??

change of limits :
x=0-> tan y =0 ---> y=0
x=1->tan y =1 -----> y= pi/4

Answer 9

integral of sec x = ln |sec x + tan x|
now just put limits
0 to pi/4.

Answer 10

Thank you for all of the help. I'm actually headed to another class right now, but I plan to finish this up later on. Thank again for the guidance.

Answer 11

welcome :)
ask if any doubts anywhere...