Integration Help: U-Substitution for $\sqrt{9x^{2}+4}$

[tweety1234]

Homework Statement

Hi,

I am not sure if you can use a u-substitution on this integral, $$\int \frac{1}{\sqrt{9x^{2}+4}} dx$$?

$$\sqrt{9x^{2}+4}$$

Can I?

I have tried it, but get an even more complicated integral from what I started with.

 

[n.karthick]

You can refer to the standard integrals in the PF library to get the solution directly.

 

[tweety1234]

You can refer to the standard integrals in the PF library to get the solution directly.

Thank you, but the table does not show me a method, and I would really like to know how to go about this.


I am right in saying we can use a substitution?

 

[Gib Z]

You can use a substitution, but not the one you used. Have you learned the theory and method of trigonometric substitutions?

 

[n.karthick]

Could you think of using the trigonometric relation $sin^2x+cos^2x=1$ to simplify your integral.

 

[HallsofIvy]

If that isn't a sufficient hint note that dividing both sides of $sin^2 x+ cos^2 x= 1$ by $cos^2 x$ gives $tan^2 x+ 1= sec^2 x$ and that
$$\sqrt{9x^2+ 4}= \sqrt{\frac{1}{4}\left(\frac{9x^2}{4}+ 1\right)}= \frac{1}{2}\sqrt{\left(\frac{3x}{2}\right)^2+ 1}$$