- #1

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I'm stuck at this one part, but i'll show you my steps so you guys can see if i'm doing it right or not.

factor out the 9...

[tex]\int \frac{sqrt(9(x^2-4/9)}{x}dx[/tex]

x = 2/3*sec(theta)

dx = 2/3*sec(theta)tan(theta) d(theta)

then i just subbed 2/3*sec(theta) for one of the X first.

[tex]sqrt(9(4/9*sec(theta)^2 - 4/9))[/tex] => [tex]sqrt(4*sec(theta)^2-4[/tex]

=> [tex]sqrt(4(sec(theta)^2-1))[/tex]

using trig. id.....

=> 2tan(theta)

now i will sub in for the other x and dx.

[tex]\int \frac{2*tan(theta)}{2/3*sec(theta)}*2/3*sec(theta)*tan(theta)[/tex]

2/3*sec(theta) canceles out each other so...

[tex]2\int tan(theta)^2[/tex]

=> trig id... [tex]2\int sec(theta)^2-1[/tex]

well i know the integral of sec(theta)^2 is just tan(theta). but what is the integral of 1? it would be x in other cases, but what would the integral of 1 be in this case?