integration

this is pretty hard: $\int\sqrt{(a^2) + ((2bx)^2)} dx$

and I have no idea...

For an "undergraduate-level" question, this is not all that difficult. It is solved quickly using basic Core 3/4 knowledge.

Anyway, there are a variety of ways you can do this.

Firstly I would consider writing it in a simpler form:


You essentially now must evaluate:


Is there any kind of substitution that may be useful here?


Alternatively, are there any theorems you can use?


Alternatively again, dependant on which form you require your final answer, is there some way to get rid of the square root that would make this more approachable?


etc.. etc..

Hope this helps!

You do realise that most of your post contradicts this

No it doesn't.

It is solved easily used Core 3/4 knowledge as the two simplest methods are

The other methods are alternatives but that doesn't change the fact that it can be solved using the less-sophisticated methods...

this is pretty hard: $\int\sqrt{(a^2) + ((2bx)^2)} dx$

and I have no idea...

I'll note that the (simpler) integral

$\dfrac{1}{\sqrt{x^2 +2bx + c}}$ was the topic of a STEP I question (2008), and the examiners gave a pretty massive hint about the form of the solution.

IMHO it seems ridiculous to say this one is easily done using C3/C4 knowledge.

You suggest that this can be solved quickly using C3/4 knowledge

The substitution of atanu requires use of reduction which is not in C3/4

IMHO it seems ridiculous to say this one is easily done using C3/C4 knowledge.

this is pretty hard: $\int\sqrt{(a^2) + ((2bx)^2)} dx$

and I have no idea...

...where the last result is part of the same formula booklet given to C3/C4 students. It can also be obtained by algebraic rearrangement.

what happened to the x

Have you done integration by substitution?
Let $u=(a^2) + ((2bx)^2)$
Differentiate $\frac{du}{dx} = 0 + (4bx)(2b) = 8b^2 x$
Make dx the subject $dx= \frac{du}{8b^2 x}$
Substitute u into the the integral and substitute dx too $\int\ (u^{1/2}) (\frac{du}{8b^2 x})$
$\int\ (u^{1/2}) (\frac{du}{8b^2 x}) = \frac{1.5(u^{1.5})}{8b^2 x} + c = \frac{1.5( ((a^2) + ((2bx)^2) ))^{1.5}}{8b^2 x} + c$