Question

let x = 2tan(theta) to evaluate the integral of: 1/((x^(2)sqrt(x^(2)+4))

Answer 1

no one did this yet?

Answer 2

nope lol is it hard or easy for u?

Answer 3

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

Answer 4

it tells you what substitution they want you to use

Answer 5

\[x=2 \tan(\theta) => dx =2 \sec^2(\theta) d \theta\]

Answer 6

\[\int\limits_{}^{}\frac{1}{(2\tan(\theta))^2 \sqrt{(2\tan(\theta))^2+4}} 2 \sec^2( \theta) d \theta\]

Answer 7

idea is to replace x by
\[2\tan(\theta)\] then use the trig identity
\[\tan^2(\theta)+1=\sec^2(\theta)\] so clear the radical.

Answer 8

that is the idea. myininaya is writing it out nicely

Answer 9

i see, but where do i go from there? i am still confused
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Answer 10

\int\limits_{}^{}\frac{1}{4\tan^2(\theta) \sqrt{4 \tan^2(\theta)+4}} 2 \sec^2(\theta) d \theta

\int\limits_{}^{}\frac{1}{\4tan^2(\theta) \sqrt{4}\sqrt{\tan^2(\theta)+1}}2 \sec^2(\theta) d \theta

\[\frac{1}{4 \sqrt{4}} \int\limits_{}^{} \frac{1}{\tan^2(\theta) \sqrt{\sec^2(\theta)}} 2 \sec^2(\theta) d \theta\]

Answer 11

lol oops

Answer 12

lol

Answer 13

lol

Answer 14

\[\int\limits_{}^{}\frac{1}{4\tan^2(\theta) \sqrt{4 \tan^2(\theta)+4}} 2 \sec^2(\theta) d \theta\]

Answer 15

\[\int\limits_{}^{}\frac{1}{\4tan^2(\theta) \sqrt{4}\sqrt{\tan^2(\theta)+1}}2 \sec^2(\theta) d \theta\]

Answer 16

you can do the work and i will take the credit.

Answer 17

hey satellite (i was looking for a soln involving paramatetric eqns) and myininaya can you look at my silo question after your done helping here thx

Answer 18

\[\int\limits\limits_{}^{}\frac{1}{4\tan^2(\theta) \sqrt{4 \tan^2(\theta)+4}} 2 \sec^2(\theta) d \theta \]

\[\int\limits\limits_{}^{}\frac{1}{4 \tan^2(\theta) \sqrt{4}\sqrt{\tan^2(\theta)+1}}2 \sec^2(\theta) d \theta \]

\[\frac{1}{4 \sqrt{4}} \int\limits\limits_{}^{} \frac{1}{\tan^2(\theta) \sqrt{\sec^2(\theta)}} 2 \sec^2(\theta) d \theta \]

Answer 19

\[\frac{2}{4 (2)} \int\limits_{}^{}\frac{\sec^2(\theta)}{\tan^2(\theta) \sec(\theta)} d \theta\]

Answer 20

\[\frac{ \not{2}}{4 (\not{2})}\int\limits_{}^{}\frac{\sec(\theta)}{\tan^2(\theta)} d \theta\]

Answer 21

\[\frac{1}{4}\int\csc(\theta)\cot(\theta)d\theta=-\frac{1}{4}\csc(\theta)\]

Answer 22

and now you have to go back...

Answer 23

\[\frac{1}{4}\int\limits_{}^{}\frac{1}{\cos(\theta)} \cdot \frac{\cos^2(\theta)}{\sin^2(\theta)} d \theta\]

Answer 24

\[\theta =tan^{-1}(\frac{x}{2})\]

Answer 25

\[\frac{1}{4} \int\limits_{}^{}\frac{\cos(\theta)}{\sin^2(\theta)} d \theta\]

Answer 26

let \[u=\sin(\theta) => du=\cos(\theta) d \theta \]

\[\frac{1}{4} \int\limits_{}^{}\frac{1}{u^2} du\]

Answer 27

\[\frac{1}{4} \int\limits_{}^{}u^{-2} du= \frac{1}{4} \cdot \frac{u^{-2+1}}{-2+1} +C\]

Answer 28

\[\frac{1}{4} \cdot \frac{u^{-1}}{-1}+C=\frac{-1}{4u}+C=\frac{-1}{4\sin(\theta)}+C\]

Answer 29

so we need to know what \[\sin(\theta) \] means in terms of x
this is where we draw a right triangle because geometry is a blast!