Integrating tan^2(u)(sec(u))du: Is There a Clean Way?

[daveed]

whats the integral of tan^2(u)(sec(u))du?

i was trying to integrate
(x^2)/sqrt(x^2+1)dx, and came into that. it turns out pretty messy though, is there a clean way to do it?

 

[shmoe]

Hi, I'm usually inclined to convert an integral like that to one with only sin's and cos's:

$$\int\frac{\sin(u)^2}{\cos(u)^3}du$$

Integration by parts will work on this if you break it up properly.

 

[dextercioby]

$$\int \frac{x^{2}}{\sqrt{x^{2}+1}} dx =...?$$
Make the natural substitution:$x\rightarrow \sinh y$

U'll be gettin' $$\int \sinh^{2}y dy$$ (1)
Consider the "sister integral" $$\int \cosh^{2}y dy$$ (2)

Consider the two expressions obtained by:(2)+(1);(2)-(1).The two new integrals will be trivials since u can use the 2 formulae from hyperbolic trigonometry:
$$\cosh^{2}y-\sinh^{2}y=1;\cosh^{2}y+\sinh^{2}y= \cosh{2y}$$

In the end u can extract this integral $$\int \sinh^{2}y dy$$ easily and then in the final result u'll have to make the substitution back
$$y\rightarrow \arg\sinh x$$

Daniel.