1. ## Integration of product

How do I integrate $\displaystyle \int{x\sqrt{4x^2+1}}$

I've tried parts with $\displaystyle x=dv$ and $\displaystyle \sqrt{4x^2+1}=u$ and vice-versa and I still don't get the answer. Help please.

N.B. When hinting at the answer, note that I cannot use hyperbolic trig functions, it's off the syllabus.

2. Originally Posted by I-Think
How do I integrate $\displaystyle \int{x\sqrt{4x^2+1}}$
This is $\displaystyle \int {\frac{{\sqrt u du}}{8}}$

3. Originally Posted by I-Think
How do I integrate $\displaystyle \int{x\sqrt{4x^2+1}}$

I've tried parts with $\displaystyle x=dv$ and $\displaystyle \sqrt{4x^2+1}=u$ and vice-versa and I still don't get the answer. Help please.

N.B. When hinting at the answer, note that I cannot use hyperbolic trig functions, it's off the syllabus.
You don't need to use integration by parts here. Consider the substitution $\displaystyle u = 4x^2$ which gives $\displaystyle dx = \frac{du}{8x}$

It should be apparent now what to do.