# Math Help - integral

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?

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

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

$x = 2 tan \theta$

$dx = 2 sec^2 \theta$

and

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

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

= $2sec\theta$

substitute back in to the original equation..

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

cross multiply

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

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

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

$x = 2\tan{t}$

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

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

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

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

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.