Here's the question you clicked on:
Kelumptus
Is the following a correct solution?
\[\int\limits_{}^{}\frac{ 1 }{ 4+x ^{2} }dx=\frac{ (x-\sqrt{4}i) }{ (x+\sqrt{4}i) }\] \[=\int\limits_{}^{}\frac{u-2\sqrt{4}i}{u}=\int\limits_{}^{}\frac{-2\sqrt{4}i}{u}=-2\sqrt{4}i \ln|u| + c\] \[=-2\sqrt{4}i \ln|x+\sqrt{4}i| + c\]
i didnt get your solution/or the question..please clarify..
The question is: is \[-2\sqrt{4}i \ln |x+\sqrt{4}i| + c\] the indefinite integral of \[\int\limits_{}^{}\frac{1}{4+x ^{2}} dx\]
you can see 1/ 4+x^2 = 1/ (x+ 2i)(x-2i) = 1/4i * (x+2i - (x-2i))/(x+2i)(x-2i) now separate(simplify) and integrate..hope that helped.. am sorry i dint get your soln much.
|dw:1354951402025:dw|
That is correct also, but i am wondering if the solution involving complex numbers is also correct.
Actually, the solution that you typed should be 1/2 Tan(x/2) + K
Ahh crap, sorry, i didn't read your solution properly, i just assumed that you were using trig substitution. Forget what i just said, let me take a closer look =)
yes, I can show that both functions has a same power series, and it shows the equality is corect.
for example tan|dw:1354952178836:dw|
for example tan^-1(i)=inf
Ahh, thats using the power series. Cheers =)