Question

Help with an indefinite integral:
int frac{ dx }{ x(x+4)(sqrt{4-x^2)} }

So, i've tried trig substitution, but i get stuck with some int (dx/(sinx+2). I think is possible to get rid of the radical and do somethig like partial fractions.
Any idea?

Answer 1

\[\int\limits \frac{ dx }{ x(x+4)(\sqrt{4-x^2)} }\]

Answer 2

any ideas yet?

Answer 3

trig sub? can you think of any?

Answer 4

x=2sin(u)

Answer 5

oh you did trig sub
you got
\[1/4 \int \tan u\frac{du}{\cos u(\sin u+2)}\]

Answer 6

cotu not tanu *

Answer 7

\[1/4\int \frac{du}{\sin u(\sin u+2)}\]

Answer 8

is that what you had?

Answer 9

you may try `t = tan(u/2)` for evaluating the `integral 1/(sin(u)+2)`

Answer 10

\[1/4 \int\limits du/2\sin(u) - 1/8 \int\limits du/(\sin(u)+2)\]

Answer 11

The trigo sub sounded like a good idea. Setting \(x=2\sin u\) to get the differential \(dx=2\cos u\,du\), you have
\[\begin{align*}\int\frac{dx}{x(x+4)\sqrt{4-x^2}}&=\int\frac{2\cos u}{2\sin u(2\sin u+4)\sqrt{4-4\sin^2u}}\,du\\\\
&=\frac{1}{4}\int\frac{du}{\sin u(\sin u+2)}\\\\
&=\frac{1}{4}\int\left(\frac{f(u)(\sin u+2)}{\sin u(\sin u+2)}+\frac{g(u)\sin u}{\sin u(\sin u+2)}\right)\,du
\end{align*}\]
Let's try finding \(f(u)\) and \(g(u)\) such that the numerator is equivalent to \(1\). If we make \(f(u)=\dfrac{1}{2}\) and \(g(u)=-\dfrac{1}{2}\), that should do the trick.
\[\frac{1}{2}\sin u+1-\frac{1}{2}\sin u=1\]
From there, rational's suggestion regarding the half-angle tangent substitution works wonders.

Answer 12

...which you've apparently figured out on your own :P

Answer 13

try rational's sub

Answer 14

hmm... well, how can i express sin(u) in terms of t?

Answer 15

\(t=\tan(u/2)\)

use double angle identity of \(\sin(u)\) in terms of \(\tan(u/2)\) :
\[\sin(u) = \frac{2\tan(u/2)}{1+\tan^2(u/2)}=\frac{2t}{1+t^2}\]

Answer 16

u will have to use long division with this one after this part

Answer 17

i think we end up with 1/(t^2+t+1)

factoring the denominator using quadratic formula,
then partial fractions...

really a very long path

Answer 18

it should be an easier way. Anyway.
Thanks a lot.

Answer 19

np:)