1. ## double-parted integral

Hello! i'm having some trouble with this question(attached file). Can someone explain the steps to remove the..foreign variable. thank you in advance

2. Originally Posted by Corum
Hello! i'm having some trouble with this question(attached file). Can someone explain the steps to remove the..foreign variable. thank you in advance
Is this really the whole question?

3. Yes. Calculate: *whats there*
seeing as it's in that interval, i suppose calculating the numeric value in t, which then multiplies the rest. the problem is that the upper limit is x...

4. Originally Posted by Corum
Yes. Calculate: *whats there*
seeing as it's in that interval, i suppose calculating the numeric value in t, which then multiplies the rest. the problem is that the upper limit is x...
No, the problem is that $\int\frac{\cos(t)}{t}dt$ is not expressible in elementary terms.

5. Ci(t). But if the limit was [0, PI] for ex.

d/dx ( $\int ln (t) dt$) = ln x

Along these lines?

6. Originally Posted by Corum
Ci(t). But if the limit was [0, PI] for ex.

d/dx ( $\int ln (t) dt$) = ln x

Along these lines?
I have no idea what you are talking about.

7. An example of the Fundamental theorem of calculus

8. Originally Posted by Corum
An example of the Fundamental theorem of calculus
There is no differential operator here. How could you possibly hope to apply the FTC?

9. Not along those lines then. Thanks! Any other suggestion how to approach it?

10. Originally Posted by Corum
Not along those lines then. Thanks! Any other suggestion how to approach it?
I don't any see possible way to do this. Maybe I am missing some little trick or something, but I have a feeling that this either isn't the whole problem or you wrote it wrong.

11. sorry there, but i'm sure i didnīt make a mistake.
heres the whole exercice:

12. Originally Posted by Drexel28
I don't any see possible way to do this. Maybe I am missing some little trick or something, but I have a feeling that this either isn't the whole problem or you wrote it wrong.
Maybe let $u=\int_1^x\frac{\cos t}{t}\,dt\implies \,du=\frac{\cos x}{x}\,dx$

So $\int\frac{\cos x}{x}\left(\int_1^x\frac{\cos t}{t}\,dt\right)^4\,dx\xrightarrow{u=\int_1^x\frac {\cos t}{t}\,dt}{}\int u^4\,du$

Is this what is needed?

13. Originally Posted by Chris L T521
Maybe let $u=\int_1^x\frac{\cos t}{t}\,dt\implies \,du=\frac{\cos x}{x}\,dx$

So $\int\frac{\cos x}{x}\left(\int_1^x\frac{\cos t}{t}\,dt\right)^4\,dx\xrightarrow{u=\int_1^x\frac {\cos t}{t}\,dt}{}\int u^4\,du$

Is this what is needed?
But you will still have to compute the integral at the end. I guess that makes sense. Good-eye!

14. Originally Posted by Drexel28
But you will still have to compute the integral at the end. I guess that makes sense. Good-eye!
True..but at least it reduces it...to something that can't be done in terms of elementary functions...

15. Yes, I've solved it now. That form is the excepted. Thanks for the help =)