Trig substitition-lost in identities

[phantomcow2]

Trig substitition--lost in identities

Homework Statement

$$\int$$$$\frac{dx}{x^{2}\sqrt{x^{2}+1}}$$

Homework Equations

It's pretty obvious that this is a trig substitution problem requiring use of tangent.

The Attempt at a Solution

$$x=tan\theta$$
$$dx=sec^{2}\theta$$
$$x^{2}=tan^{2}\theta$$

Substitute it in.

$$\frac{sec^{2}\theta}{tan^{2}\theta\sqrt{tan^{2}\theta + 1}}$$

But the $$\sqrt{tan^{2}\theta + 1}$$ simplifies to $$sec\theta$$

Now before I integrate, I need to simplify. The obvious simplification is the $$sec^{2}\theta$$ and $$sec\theta$$ in the denominator, leaving me with
$$\frac{sec\theta}{tan^{2}\theta}$$

This is starting to look fishy to me. I think I've begun to develop an instinct telling me when I am doing something incorrectly. I simplify this to $$\frac{cos\theta}{sin^{2}\theta}$$

Now I need to ingrate this, but it doesn't look promising. Can someone tell me where I went wrong?

 

[Avodyne]

Nothing wrong so far. Time for another substitution.

 

[phantomcow2]

Oh, haha, you're right! I need to start thinking to substitute more often :p

u= sin and du = cos

Yep, straight forward from here. Thanks so much.