How would i integrate this.

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April 21st 2010, 06:35 AM
x5pyd3rx

How would i integrate this.

$\frac{cosx}{sinx} * cos x dx$

Im so lost that second cos x is really thorwing me off
April 21st 2010, 06:41 AM
Deadstar

My initial guess would be to simplify it to...
$<br /> \frac{\cos^2(x)}{\sin(x)} = \frac{1-\sin^2(x)}{\sin(x)} = \frac{1}{\sin(x)} - \sin(x)$

But integrating 1/sin(x) is not exactly trivial.

Will have another think
April 21st 2010, 06:48 AM
Deadstar

Nah that's still the best I can come up with.

$\int \frac{1}{\sin(x)} dx = \int \frac{\sin(x)}{\sin^2(x)}dx = \frac{\sin(x)}{1-\cos^2(x)} dx$ .

Let $u = \cos(x)$ , $du = \sin(x)dx$

Hence integral becomes...

$\int \frac{1}{1-u^2}du = arctanh(u)$
April 21st 2010, 06:48 AM
Soroban

Hello, x5pyd3rx!

Quote:

$\int \frac{\cos x}{\sin x}\cdot\cos x\, dx$

$\int\frac{\cos^2\!x}{\sin x}\,dx \;\;=\;\;\int\frac{1-\sin^2\!x}{\sin x}\,dx \;\;=\;\;\int\left(\frac{1}{\sin x} - \sin x\right)\,dx$

. . . . . . . . $=\;\int(\csc x - \sin x)\,dx \;\;=\;\;\ln|\csc x - \cot x| + \cos x + C$
April 21st 2010, 06:51 AM
Deadstar

Quote:

Originally Posted by Soroban

Hello, x5pyd3rx!

$\int\frac{\cos^2\!x}{\sin x}\,dx \;\;=\;\;\int\frac{1-\sin^2\!x}{\sin x}\,dx \;\;=\;\;\int\left(\frac{1}{\sin x} - \sin x\right)\,dx$

. . . . . . . . $=\;\int(\csc x - \sin x)\,dx \;\;=\;\;\ln|\csc x - \cot x| + \cos x + C$

That.

Is easier.

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