# Thread: integration of 3 variables

1. ## integration of 3 variables

integral of <||cos t, sin t, t||> dt

the answer in the back has a natural log and i don't get it when i work it out???

2. Hello,

Originally Posted by chris25
integral of <||cos t, sin t, t||> dt

the answer in the back has a natural log and i don't get it when i work it out???
Is it ||...|| for the norm in basis 2 ?

It would be $||\cos t, \sin t, t||=\sqrt{\cos^2t+\sin^2t+t^2}=\sqrt{1+t^2}$

3. no it means the length

4. Originally Posted by chris25
no it means the length
Well, it has to be the same...

Can you state and write the exact question ?
Are there really < and > ?

5. evaluate the indefinite integral...

S||(cos t i + sin t j + t k )|| dt

6. Originally Posted by chris25
evaluate the indefinite integral...

S||(cos t i + sin t j + t k )|| dt
So yeah, you have to calculate $\int \sqrt{1+t^2} dt$

But I'm not sure how to calculate it...

7. Originally Posted by Moo
So yeah, you have to calculate $\int \sqrt{1+t^2} dt$

But I'm not sure how to calculate it...
$\int\sqrt{1+x^2}dx$

let $x=\tan(\theta)$

so $dx=\sec^2(\theta)d\theta$

Giving us after simplification

$\int\sec^3(\theta)d\theta$

Which by a manipulation and double parts gives http://www.mathhelpforum.com/math-he...-integral.html

$\frac{1}{2}\bigg[\ln|\sec(\theta)+\tan(\theta)|+\tan(\theta)\sec(\t heta)\bigg]$

Now back subbing in that $\theta=\arctan(x)$

we get

$\frac{1}{2}\bigg[\ln|\sqrt{x^2+1}+x|+x\sqrt{x^2+1}\bigg]$

8. Originally Posted by Mathstud28
$\int\sqrt{1+x^2}dx$

let $x=\tan(\theta)$

so $dx=\sec^2(\theta)d\theta$

Giving us after simplification

$\int\sec^3(\theta)d\theta$

Which by a manipulation and double parts gives http://www.mathhelpforum.com/math-he...-integral.html

$\frac{1}{2}\bigg[\ln|\sec(\theta)+\tan(\theta)|+\tan(\theta)\sec(\t heta)\bigg]$

Now back subbing in that $\theta=\arctan(x)$

we get

$\frac{1}{2}\bigg[\ln|\sqrt{x^2+1}+x|+x\sqrt{x^2+1}\bigg]$
The substitution $x = \sinh \theta$ will reduce the algebra.

9. Originally Posted by mr fantastic
The substitution $x = \sinh \theta$ will reduce the algebra.
Yeah, but I wanted to supply a more commonly known sub

But to explain what Mr. F astutely noted

Let $x=\sinh(\theta)$

So $dx=\cosh(\theta)$

Giving us

$\int\sqrt{1+\sinh^2(\theta)}\cosh(\theta)d\theta=\ int\cosh^2(\theta)d\theta$

Now making the appropriate hyperbolic identity we get

$\frac{1}{2}\bigg[\theta+\cosh(\theta)\sinh(\theta)\bigg]$

Now just back sub

10. Originally Posted by mr fantastic
The substitution $x = \sinh \theta$ will reduce the algebra.
Actually, the OP mentioned that the answer contained a logarithm