# Homework Help: Calculus BC Trigonometric Substitition

1. Nov 11, 2009

### hwmaltby

1. The problem statement, all variables and given/known data

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

2. Relevant equations

$$\sin^2{\theta}+\cos^2{\theta}=1$$

3. The attempt at a solution

$$x=\tan{\theta}$$

I simplified it down to:

$$\int \frac {1}{\sin{\theta} \cdot \cos^2{\theta}} \, d\theta$$

Which I do not know how to integrate.

Any help would be wonderful!

Last edited: Nov 11, 2009
2. Nov 11, 2009

### hwmaltby

Never mind... I rewrote it as:

$$\int \csc{\theta}\cdot\sec^2{\theta} \, d\theta$$

And used u-substition:

$$u=\sec{\theta}$$

$$\int \frac{\csc{\theta} \cdot \sec^2{\theta}}{\tan{\theta} \cdot \sec{\theta}} \, du$$

$$\int \csc^2{\theta} \, du$$

$$\int \frac{u^2}{u^2-1} \, du$$

$$\frac{1}{2} \cdot \ln |\frac{u-1}{u+1}| +u+C$$

I then simplified using my previous substitutions and logarithm rules to get my final answer:

$$\ln |\frac{\sqrt{1+x^2}-1}{x}|+\sqrt{1+x^2}+C$$

3. Nov 11, 2009

### Staff: Mentor

And you can check by differentiating your answer. If you end up with sqrt(1 + x^2)/x, you're golden.