Integration technique: Multiplication by a form of 1.

[Zeth]

I would much appreciate if someone could direct me to a webpage that has examples of this. My book, Thomas Calculus, only has one example (elementary) and this is for self study so I don't have lecture notes to go with it. All the questions in it are on trig integrals if that's any help.

 

[morson]

Take the simple case $$\int \frac{d\theta}{cos(\theta)}\$$

Multiply the integrand by $$\frac{cos(\theta)}{cos(\theta)}\$$, which is 1, and you get:

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

Then, if you put $$u = sin(\theta)$$, and make all necessary substitutions, the integral becomes:

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

Which is trivial to integrate by partial fractions. So hopefully I have shown that multiplication by a form of 1 can be useful in finding integrals that look intimidating such as the integral of sec(x), as above.

 

[Zeth]

Oh I do know it is useful, that's why I want the practice. It's just that I wanted to get more of a feel for how it's done by some examples that are worked through. Worst comes to worst I'll just make some up and back differentiate them and see what happens then.