Integrate sqrt(1+4x^2) using u=2x+1?

[Swatch]

I am trying to integrate sqrt(1+4*x^2)
I have been trying to rewrite this into sqrt(-4x+(2x + 1)^2) and putting u=2x+1 and substituting but I don't think that makes this any easier. Could someone please give me a hint.

 

[marlon]

I am trying to integrate sqrt(1+4*x^2)
I have been trying to rewrite this into sqrt(-4x+(2x + 1)^2) and putting u=2x+1 and substituting but I don't think that makes this any easier. Could someone please give me a hint.

The answer is Partial Integration...Give it a try and let me know

marlon

edit hint : do the partial integration right a way with the sqrt(1+4x^2)dx. After this you will need to apply the (+1 -1) trick in the integrand's numerator. If you want you can first do the substitution u=2x to get rid of the 4, but it is not compulsory...

 

[Swatch]

After some hard work and a lot of eraser I got the right answer. Thank you marlon.

 

[amcavoy]

Also:

$$\int\sqrt{1+\left(2x\right)^2}\,dx=\frac{1}{2}\int\cosh^2{x}\,dx$$

Then use the identity for the double angle to simplify that integral.

 

[benorin]

Try $$x=\frac{1}{2}\tan \theta \Rightarrow \sqrt{1+4x^2}=\sec\theta$$.