integration question

**needhelp101** · January 19th 2009, 07:35 PM

right now i'm stuck on this integration problem. it's integrating (1/theta^2)cos(1/theta). any suggestions??

**Soroban** · January 19th 2009, 07:55 PM

Hello, needhelp101!

$\int \frac{\cos\left(\frac{1}{\theta}\right)}{\theta^2} \,d\theta$

We have: . $\int \cos\left(\theta^{-1}\right)\,\left(\theta^{-2}d\theta\right)$

Let $u \:=\:\theta^{-1} \quad\Rightarrow\quad du \:=\:-\theta^{-2}d\theta \quad\Rightarrow\quad \theta^{-2}d\theta \:=\:-du$

Substitute: . $\int \cos u\,(-du) \;=\;-\!\!\int\cos u\,du$

Got it?

**needhelp101** · January 19th 2009, 08:00 PM

yup. i got it now. thanks for the help. i really appreciate it.

**hatsoff** · January 19th 2009, 08:06 PM

Originally Posted by needhelp101

right now i'm stuck on this integration problem. it's integrating (1/theta^2)cos(1/theta). any suggestions??

$\int\theta^{-2}\cos{\theta^{-1}}d\theta$

Substitution:

$u=\theta^{-1}$

$-du=\theta^{-2}d\theta$

$\int\theta^{-2}\cos{\theta^{-1}}d\theta=-\int\cos{u}\;du$

$=-\sin{u}$

$=-\sin{\theta^{-1}}$

Check the solution by taking the derivative:

$-\frac{d}{d\theta}\sin{\theta^{-1}}=\theta^{-2}\cos{\theta^{-1}}$

Looks good!

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