double-parted integral

**Corum** · January 6th 2010, 05:08 PM

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

**Drexel28** · January 6th 2010, 05:12 PM

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?

**Corum** · January 6th 2010, 05:16 PM

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...

**Drexel28** · January 6th 2010, 05:18 PM

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.

**Corum** · January 6th 2010, 05:23 PM

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

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

Along these lines?

**Drexel28** · January 6th 2010, 05:42 PM

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.

**Corum** · January 6th 2010, 05:45 PM

An example of the Fundamental theorem of calculus

**Drexel28** · January 6th 2010, 05:49 PM

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?

**Corum** · January 6th 2010, 05:52 PM

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

**Drexel28** · January 6th 2010, 06:03 PM

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.

**Corum** · January 6th 2010, 06:11 PM

sorry there, but i'm sure i didn´t make a mistake.
heres the whole exercice:

**Chris L T521** · January 6th 2010, 06:56 PM

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?

**Drexel28** · January 6th 2010, 06:59 PM

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!

**Chris L T521** · January 6th 2010, 07:00 PM

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...

**Corum** · January 8th 2010, 02:08 AM

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

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