Help integrating complex function!

**platinumpimp68plus1** · February 8th 2010, 08:23 PM

I'm stuck... its been awhile since I've done integrals.

So the integral is $\int\frac{dz}{z}$ along the straight line segment from 1 to i. I parameterized the curve as $\gamma:[0,1]\rightarrow C$ , defined as $\gamma(t)=(-t+1)+it$ and then used the property that dividing 1 by z is the equivalent of $z^{-1}$ . So I have (with the constant (-1+i) omitted):

$)\int(\frac{-t+1}{(-t+1)^2+t^2})-i\frac{t}{(-t+1)^2+t^2}dt\\=\int(\frac{-t+1}{(-t+1)^2+t^2})dt-i\int\frac{t}{(-t+1)^2+t^2}dt$

I have no clue how to integrate these... is there some trick I'm forgetting? any help?

**Drexel28** · February 8th 2010, 08:26 PM

Originally Posted by platinumpimp68plus1

I'm stuck... its been awhile since I've done integrals.

So the integral is $\int\frac{dz}{z}$ along the straight line segment from 1 to i. I parameterized the curve as $\gamma:[0,1]\rightarrow C$ , defined as $\gamma(t)=(-t+1)+it$ and then used the property that dividing 1 by z is the equivalent of $z^{-1}$ . So I have:

$\int(\frac{-t+1}{(-t+1)^2+t^2})-i\frac{t}{(-t+1)^2+t^2}dt\\=\int(\frac{-t+1}{(-t+1)^2+t^2})dt-i\int\frac{t}{(-t+1)^2+t^2}dt$

I have no clue how to integrate these... is there some trick I'm forgetting? any help?

$\int\frac{-t+1}{(-t+1)^2+t^2}dt=\int\frac{-t+1}{t^2-2t+1+t^2}dt=\int\frac{-t+1}{2t^2-2t+1}dt$ . Let $z=2t^2-2t+1\implies dz=2t-2$ ..

**platinumpimp68plus1** · February 8th 2010, 08:59 PM

Originally Posted by Drexel28

$\int\frac{-t+1}{(-t+1)^2+t^2}dt=\int\frac{-t+1}{t^2-2t+1+t^2}dt=\int\frac{-t+1}{2t^2-2t+1}dt$ . Let $z=2t^2-2t+1\implies dz=2t-2$ ..

I got kind of excited until I noticed its actually $dz=4t-2$ ... but that lead me on this train of thought:

(Integrating the real term):

$z=2t^2-2t+1 \implies dz=4t-2$
$-t+1= \frac{-1}{4} (4t-4)=\frac{-1}{4} (4t-2-2)=\frac{-1}{4} (4t-2)+\frac{1}{2}$

So:

$=\frac{-1}{4}\int\frac{4t-2}{2t^2-2t+1}dt+\frac{1}{2}\int<br /> \frac{1}{2t^2-2t+1}dt$

Which should be...

$=-\frac{1}{4}\ln(2t^2-2t+1)+\frac{1}{2}\tan^-1(2x-1)$

yay/nay?

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