Question

Evaluate the Integral:

Integral( dx/(x*Sqrt(36x^(2)-2))
I also attached the sheet with the answer choices. I got B but im not sure if its correct...

Answer 1

The integral is: Integral( dx/(x*Sqrt(36x^(2)-2))
I also attached the sheet with the answer choices. I got B but im not sure if its correct...

Answer 2

\[\int\frac{dx}{x\sqrt{36x^2-2}}\\
\text{Let }x=\frac{\sqrt2}{6}\sec t\\
dx=\frac{\sqrt2}{6}\sec t\tan t\; dt\\

\int\frac{\frac{\sqrt2}{6}\sec t\tan t}{\frac{\sqrt2}{6}\sec t\sqrt{36\left(\frac{\sqrt2}{6}\sec t\right)^2-2}}dt\\
\int\frac{\tan t}{\sqrt{36\left(\frac{2}{36}\sec^2t\right)-2}}dt\\
\int\frac{\tan t}{\sqrt{2\sec^2t-2}}dt\\
\frac{1}{\sqrt2}\int\frac{\tan t}{\sqrt{\sec^2t-1}}dt\\
\frac{1}{\sqrt2}\int\frac{\tan t}{\sqrt{\tan^2t}}dt\\
\frac{1}{\sqrt2}\int\frac{\tan t}{\tan t}dt\\
\frac{1}{\sqrt2}\int dt\\
\frac{1}{\sqrt2}t+C\\
\text{Note: }x=\frac{\sqrt2}{6}\sec t\iff t=\sec^{-1}\left(\frac{6}{\sqrt2}\right)\\
\frac{1}{\sqrt2}\sec^{-1}\left(\frac{6}{\sqrt2}\right)+C\\
\frac{\sqrt2}{2}\sec^{-1}(3\sqrt2)+C\]

Answer 3

I think you're right with choosing B, but the answer might be C. I'm not sure when the absolute value bars are necessary.

Answer 4

The last two lines are missing an x... Should be
\[\frac{\sqrt2}{2}\sec^{-1}(3\sqrt2x)+C, \text{ or possibly}\\
\frac{\sqrt2}{2}\sec^{-1}|3\sqrt2x|+C\]

Answer 5

Darn... so the answer would be C then for sure? Idk how I got sin anyway...

Answer 6

Oh, I thought B contained a secant, too. Yes, the answer must be C.

Answer 7

Alright thanks for your help.

Answer 8

You're welcome.