1. ## integral

I have a problem i dont have the solution manual too and i am stumped on how to complete.

This could be a trigonometric substitution?

$\displaystyle \int_1^2 \frac {dx}{x \sqrt {4 + x^2}}$

this is what i came up with but i dont think its right..

$\displaystyle x = 2 tan \theta$

$\displaystyle dx = 2 sec^2 \theta$

and

$\displaystyle \sqrt {2^2 + ( 2tan \theta)2}}$

identity
$\displaystyle (tan^2\theta -1)2^2$

=$\displaystyle 2sec\theta$

substitute back in to the original equation..

$\displaystyle \int_1^2 \frac{2 sec^2 \theta} {2 tan\theta 2 sec\theta}2 sec^2\theta$

cross multiply

$\displaystyle \int_1^2 \frac{2 sec^2 \theta} {2 tan\theta}$

not sure if the 2 can cross out or what to do with them or if i am even right at this point?

2. ## Re: integral

$\displaystyle \int_1^2 \frac{dx}{x\sqrt{4+x^2}}$

$\displaystyle x = 2\tan{t}$

$\displaystyle dx = 2\sec^2{t} \, dt$

$\displaystyle \int_{\arctan(1/2)}^{\pi /4} \frac{2\sec^2{t}}{2\tan{t} \cdot 2\sec{t}} \, dt$

$\displaystyle \frac{1}{2} \int_{\arctan(1/2)}^{\pi /4} \frac{\sec{t}}{\tan{t}} \, dt$

$\displaystyle \frac{1}{2} \int_{\arctan(1/2)}^{\pi /4} \csc{t} \, dt$

finish it?

3. ## Re: integral

where did you get the arctan and pi from?

4. ## Re: integral

When you do a substitution, you must change the limits of integration.
Some find it easier to just leave the limits blank until back-substituting towards the end.
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