Integration using Trig. Substitution

[3.141592654]

Homework Statement

$$\int\sqrt{X^2+1}dX$$

Homework Equations

The Attempt at a Solution

I used the substitution X=tan $$\theta$$

So, $$dX=(sec^2 \theta$$) d$$\theta$$

Substituting in for X, I get:

$$\int\sqrt{(tan^2 \theta)+1}(sec^2 \theta) d\theta$$

= $$\int\sqrt{(sec^2 \theta)}(sec^2 \theta) d\theta$$

= $$\int(sec \theta)(sec^2 \theta) d\theta$$

I then converted secants into cosines:

= $$\int\frac{1}{(cos \theta)(cos^2 \theta)} d\theta$$

= $$\int\frac{1}{(cos \theta)(1-sin^2 \theta)} d\theta$$

I then used U-sub:

$$u=sin \theta$$

$$du=cos \theta$$

$$\frac{du}{cos \theta}=d\theta$$

= $$\int\frac{du}{(cos^2 \theta)(1-u^2)}$$

= $$\int\frac{du}{(1-u^2)(1-u^2)}$$

= $$\int\frac{du}{(1-2u^2+u^4)}$$

= $$\int\frac{du}{(u^4-2u^2+1)}$$

= $$\int\frac{du}{u^2(u^2-2)+1}$$

I see this:

= $$\int\frac{du}{[u\sqrt{u^2-2}]^2+1}$$

I was hoping I could then use substitution and have the integral of $$arctan$$, but it looks much more complicated than I thought. Am I anywhere near the right track?

 

[l'Hôpital]

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

At that point, you could attempt partial fractions. However, an easier way is here:

$$\int(sec \theta)(sec^2 \theta) d\theta = \int sec^3 \theta d\theta$$

This is a famous (sorta, anyways) integral. Use parts twice.

Alternatively, since this integral is so famous, it even has its own wikipedia page.

http://en.wikipedia.org/wiki/Integral_of_secant_cubed

I'd suggest you trying the derivation yourself before looking at the "answer" but it's there if you need it.