Question

integrate using trig sub. problem below.

Answer 1

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

Answer 2

put x=2tan u

Answer 3

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

Answer 4

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,\[\large \frac{1}{4}\int\limits\limits \frac{\sec^2\theta \; d\theta}{\tan^2\theta \sqrt{\color{royalblue}{\sec^2\theta}}}\]

Answer 5

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

Answer 6

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? :)

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

Yah I did that upside down :) woops lol

Answer 13

i think its cos over sin squared

Answer 14

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

Answer 15

okay thanks a lot!