Integral problem

uberathlete

New member
Joined
Jan 16, 2006
Messages
48
Hi everyone. I'm having problems on how to integrate 2/(x^2+4) . I know the answer is arcsin(1/2x) + C, but I can't seem to get to that answer. I tried regular substitution, and integration by parts and they don't seem to work. I'm not so sure if trig substitution would work either. Any help on this would be greatly appreciated. Thanks!
 
Surely you mean the answer is \(\displaystyle \L
\arctan \left( {\frac{x}{2}} \right) + C\)

Consider \(\displaystyle \L
\frac{2}{{x^2 + 4}} = \left( {\frac{1}{4}} \right)\frac{2}{{\left( {x/2} \right)^2 + 1}}\), then let \(\displaystyle \L
u = (x/2)\).

The arctangent form is \(\displaystyle \L
\frac{{du}}{{u^2 + 1}}\).
 
It doesn't equal sin1(x2)\displaystyle sin^{-1}(\frac{x}{2}). I believe you mean

tan1(x2)\displaystyle tan^{-1}(\frac{x}{2})


Let x=2u and dx=2du

2122+x2dx\displaystyle 2\int{\frac{1}{2^{2}+x^{2}}}dx

22du22+22u2\displaystyle 2\int{\frac{2du}{2^{2}+2^{2}u^{2}}}

2(12)du1+u2\displaystyle 2(\frac{1}{2})\int{\frac{du}{1+u^{2}}}

tan1(u)+C\displaystyle tan^{-1}(u)+C

Since u=x2\displaystyle \frac{x}{2}

tan1(x2)\displaystyle tan^{-1}(\frac{x}{2})+C
 
Hello, uberathlete!

Integrate: \(\displaystyle \L\,\int\frac{2}{x^2\,+\,4}\,dx\)

I know the answer is: arcsin(x2)+C  \displaystyle \,\arcsin\left(\frac{x}{2}\right)\,+\,C\; . . . of course, you mean arctangent
Trig Substitution is the way to go . . .

Let: x=2tanθ        dx=2sec2θdθ\displaystyle x\,=\,2\cdot\tan\theta\;\;\Rightarrow\;\;dx\,=\,2\cdot\sec^2\theta\,d\theta

And: \(\displaystyle \,x^2\,+\,4\:=\:(2\tan\theta)^2\,+\,4\:=\:4\tan^2\theta\,+\,4\:=\:4(\tan^2\theta\,+\,1)\:=\:4\cdot\sec^2\theta\)


Substitute: \(\displaystyle \L\:\int\frac{2}{4\cdot\sec^2\theta}\)\(\displaystyle \cdot(2\cdot\sec^2\theta\,d\theta)\;=\;\L\int d\theta \;= \;\theta\,+\,C\)


Back-substitute:  x=2tanθ        tanθ=x2        θ=arctan(x2)\displaystyle \;x = 2\cdot\tan\theta\;\;\Rightarrow\;\;\tan\theta\,=\,\frac{x}{2}\;\;\Rightarrow\;\;\theta\,=\,\arctan\left(\frac{x}{2}\right)


\(\displaystyle \text{Answer: }\:\L\arctan\left(\frac{x}{2}\right)\,+\,C\)
 
Top