Definite integral of rational function

[nlsherrill]

Homework Statement

The definite integral of (t^3 + t -1)/(sin(t)) from 2 to x^2

Homework Equations

The Attempt at a Solution

First off, I don't have the solution anywhere, my teacher just gave this to us to work on for the final exam review.

I can think of a few things. I know for the definite integrals you can first simplify the expression as an indefinite integral, then use the fundamental theorem to solve. I know the integral of 1/sin(t)=ln(sint), and integral of (t^3 + t -1)=(t^4/4 + t^2/2 -t) so for the indefinite integral I have..

(t^4/4 + t^2/2 -t)ln(sint). do I then just plug in the upper and lower limits to simplify? I kind of feel like I'm on the wrong track because it looks sloppy and usually he gives us problems that have reasonable looking solutions.

Another attempt is that I recognized 1/sin(t)=csc(t). So then I am left with the integral of (t^3 + t -1)(csc(t))... but I'm not sure if substitution would work.

 

[Mark44]

Homework Statement

The definite integral of (t^3 + t -1)/(sin(t)) from 2 to x^2

Homework Equations

The Attempt at a Solution

First off, I don't have the solution anywhere, my teacher just gave this to us to work on for the final exam review.

I can think of a few things. I know for the definite integrals you can first simplify the expression as an indefinite integral, then use the fundamental theorem to solve. I know the integral of 1/sin(t)=ln(sint),

Then you know something that isn't true! You seem to be thinking that

and integral of (t^3 + t -1)=(t^4/4 + t^2/2 -t) so for the indefinite integral I have..

(t^4/4 + t^2/2 -t)ln(sint).

No, that's not right, either.

do I then just plug in the upper and lower limits to simplify? I kind of feel like I'm on the wrong track because it looks sloppy and usually he gives us problems that have reasonable looking solutions.

Another attempt is that I recognized 1/sin(t)=csc(t). So then I am left with the integral of (t^3 + t -1)(csc(t))... but I'm not sure if substitution would work.

 

[Mark44]

What is the complete problem statement? I have the feeling that you have omitted an important detail.

 

[nlsherrill]

What is the complete problem statement? I have the feeling that you have omitted an important detail.

I am sorry but all I have put down is all that's on the paper. Is there more information needed to solve it?

 

[Mark44]

The integral you gave looks to be pretty difficult. I have a suspicion that the problem is something like this:

$$f(x) = \int_2^{x^2} \frac{(t^3 + t - 1)dt}{sin(t)}$$
Find f'(x).​

 

[nlsherrill]

The integral you gave looks to be pretty difficult. I have a suspicion that the problem is something like this:

$$f(x) = \int_2^{x^2} \frac{(t^3 + t - 1)dt}{sin(t)}$$
Find f'(x).​

yes that is exactly it. Sorry I don't know how to do the Latex.

This problem is way harder than the others my prof gave us for the final exam review(calculus 1). I guess its possible I wrote it down incorrectly.

 

[Mark44]

Whether you know how to use LaTeX or not, you need to give the problem statement exactly as given, which you did not do. If you don't provide the given information we are likely to waste a lot of time trying 1) to solve a different problem than the one that was given, and 2) to solve a problem that is much more difficult than it needs to be.

You omitted very important information!

To do this problem you don't need to carry out the integration, but you do need to know about the Fundamental Theorem of Calculus.