The fundamental theorem of calculus(I think;) )

[beaf123]

Homework Statement

Been doing some old exams lately and found out that something I have problems with is questions of the type ( example):
Differente the function:

∫ (x^2 ),(1), ln(t^2) dt

Sorry for the bad writing.
(x^2 ),(1), is the intgral from 1 to X^2

It should be fairly simple, but my brain somehow clicks when I see this question. Maybe because its integration AND derrivation. Could someone explain the steps they use to solve this exircise.

And would I get the same answer if I integrated the function ( maybe not the best to do in this case ), solved for the integral and then differented the integral?

Thank you!:)

 

[HallsofIvy]

So your problem is "Find the derivative of $\int_1^{x^2} ln(t^2)dt$".

You mention the "Fundamental Theorem" and I assume you mean the "Fundamental Theorem of Calculus"! One part of that says that $d/dx \int_a^x f(t)dt= f(x)$
Here, the only problem is that the upper limit is $x^2$ rather than x.

So let $u= x^2$. The integral becomes $\int_1^u ln(t^2)dt$. What is the derivative of that with respect to u? And to find the derivative with respect to x, use the chain rule: df/dx= (df/du)(du/dx).

 

[beaf123]

Im confused. Why does I have to use the chain rule? Cant I just stick X^2 in there and get: ln(X^2)^2 = ln (x)^4 = 4lnx?

;-)

 

[Robert1986]

Another way to think about it is that after doing the integration, you will have some expression like $F(x^2)-F(1)$. Now, when you take the derivative, you will have to use the chain rule (and remember $F(1)$ is a constant) so that
$\frac{d}{dx}(F(x)-F(1)) = F'(x^2)\frac{d(x^2)}{dx}$. Now, use the fact that $F'=f$ (which is the FTC you reference).

 

[Mark44]

Im confused.

Yes, very much.

Why does I have to use the chain rule? Cant I just stick X^2 in there and get: ln(X^2)^2 = ln (x)^4 = 4lnx?

It's hard to know where to start.

Are you thinking that
$$ \int ln(t^2) dt = (ln(t^2))^2?$$

To use the Fundamental Theorem of Calculus, all of the important pieces have to be exactly as in the theorem -- differentiation has to be with respect to the same variable as is present in the upper limit of integration.

For your problem, it's differentiation with respect to x, but it's x² as a limit. Reread what HallsOfIvy said you need to do.

 

[Ray Vickson]

Homework Statement

Been doing some old exams lately and found out that something I have problems with is questions of the type ( example):
Differente the function:

∫ (x^2 ),(1), ln(t^2) dt

Sorry for the bad writing.
(x^2 ),(1), is the intgral from 1 to X^2

It should be fairly simple, but my brain somehow clicks when I see this question. Maybe because its integration AND derrivation. Could someone explain the steps they use to solve this exircise.

And would I get the same answer if I integrated the function ( maybe not the best to do in this case ), solved for the integral and then differented the integral?

Thank you!:)

You can write this as ∫_{1..x^2} ln(t^2) dt, or as ∫{ln(t^2): t=1..x^2}. Anyway, do you know the general formula for
$$\frac{d}{dx} \int_1^{f(x)} F(t) \, dt?$$ If you do, just use it. Note that you can find the derivative of the integral, even if you cannot do the integral itself.

RGV

 

[beaf123]

Im going to spend some time working with this, but
if we had the same exircise but with x instead of X^2, would the answer be ln(x^2)?

 

[Mark44]

Im going to spend some time working with this, but
if we had the same exircise but with x instead of X^2, would the answer be ln(x^2)?

Yes.