# Indefinite integral approximation technique

1. Mar 19, 2010

### cdotter

1. The problem statement, all variables and given/known data
$\int_1^\infty \! \frac{(sin(x)+5)}{x^3} \, dx$
"Find two simpler integrals, one larger and one smaller."

2. Relevant equations

3. The attempt at a solution

How could I make this a simpler (ie, solvable) integral? It's been straight forward with other integrals like this:

$\int_1^\infty \! \frac{1}{\sqrt{x^4+10}} \, dx \rightarrow \int_1^\infty \! \frac{1}{\sqrt{x^4}} \, dx$

Doing this would give an overestimate since the denominator is smaller. But how would I do it for the integral in question?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 19, 2010

### Count Iblis

Can you find upper and lower bounds for sin(x)?

3. Mar 19, 2010

### cdotter

sin(infinity) and sin(1)? I don't understand what you mean.

4. Mar 19, 2010

### Char. Limit

Yes, you can. Think of what y=sin(x) is restricted to...

5. Mar 19, 2010

### cdotter

-1 and 1.

6. Mar 19, 2010

### Char. Limit

Exactly. So sin(x) is always less than or equal to 1.

You can use that.

7. Mar 19, 2010

### Staff: Mentor

And so ? <= sin(x) + 5 <= ?

Filling in the question marks should make two definitie integrals (not indefinite integrals) that are a lot easier to evaluate.