Question

help with trig substiution (writing the equation)

Answer 1

\[\int\limits (\sqrt{x^2-4})/x \]

Answer 2

x=2secTheta

Answer 3

Well we have
1-sin^2(x)=cos^2(x)
1+tan^2(x)=sec^2(x)
sec^2(x)-1=tan^2(x)

So yeah we are definitely involving sec here in our sub.

Answer 4

In yes 2sec(theta)=x to be exact

Answer 5

after integrating i got 2tan(theta) - theta

Answer 6

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

Answer 7

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

Answer 8

Is that what you had before simplifying?

Answer 9

yes

Answer 10

Ok so after simplifying and before integrating what did you have?

Answer 11

integral 2sec^2 - 1

Answer 12

theta sorry lol

Answer 13

you mean 2(sec^2(theta)-1)?

Answer 14

2tan(theta)-2theta+C

Answer 15

\[\int\limits2\sec^2\theta - 1\]

Answer 16

It should be \[\int\limits_{}^{}2(\sec^2(\theta)-1) d \theta \]

Answer 17

yeah that sorry

Answer 18

\[\int\limits_{}^{}2 \tan^2(\theta) d \theta=\int\limits_{}^{}2(\sec^2(\theta)-1)) d \theta =2(\tan(\theta)-\theta)+C \\ 2 \tan(\theta)-2 \theta +C \]

Answer 19

We need to write this in terms of x though.

Answer 20

recall x/2=sec(theta)