## anonymous 3 years ago integrate using trig sub. problem below.

1. anonymous

$\int\limits_{1/2}^{1} dx/ x ^{2}\sqrt{x ^{2} + 4}$

2. amoodarya

put x=2tan u

3. anonymous

i got as far as $1/4 \int\limits_{1/2}^{1} \sec ^{2}\theta/ \tan ^{3}\theta+\tan ^{2}\theta$ but im not sure what to do after

4. zepdrix

After you plug everything in and factor out the constants, you should have this. $\large \frac{1}{4}\int\limits \frac{\sec^2\theta \; d\theta}{\tan^2\theta \sqrt{\color{royalblue}{\tan^2\theta+1}}}$ I'm not sure what you did on the bottom.. hmm. From here we want to use an identity, $\large \color{royalblue}{\tan^2\theta+1=\sec^2\theta}$Changing the integral to,$\large \frac{1}{4}\int\limits\limits \frac{\sec^2\theta \; d\theta}{\tan^2\theta \sqrt{\color{royalblue}{\sec^2\theta}}}$

5. anonymous

okay im left with $1/4\int\limits_{1/2}^{1} \sec \theta/\sec ^{2}\theta -1$

6. zepdrix

No you're left with,$\large \frac{1}{4}\int\limits \frac{\sec \theta}{\tan^2\theta}d \theta$ You keep making weird substitutions.. I'm not sure why :c Oh you wanted it all in terms of secant I guess? :)

7. zepdrix

From here, it's probably better to convert everything to sines and cosines and the integral should be fairly simple from there :)

8. anonymous

yeah thats what i was trying to do. okay let me try it with sines and cosines

9. anonymous

still a bit confused as to what to do with the term thats squared

10. zepdrix

$\large \frac{1}{4}\int\limits \frac{\sin \theta}{\cos^2\theta}d \theta$So we get this I think? Yah it might seem a little tricky at first :) But it's just a simple U substitution. Let $$u=\cos \theta$$.

11. zepdrix

Woops did I set that up correctly? I was trying to do the sine and cosine conversion in my head... thinking.

12. zepdrix

Yah I did that upside down :) woops lol

13. anonymous

i think its cos over sin squared

14. zepdrix

yah good call XD whatever is on the bottom is your u :3

15. anonymous

okay thanks a lot!