Solving Definite Integral: x∫sqrt(5 + t^3) - e^t^2 dt

[johnhuntsman]

Doing corrections on a test and I'm trying to solve this integral and I'm having quite a bit of trouble with it:

__x
D_x∫sqrt(5 + t^3) - e^t^2 dt
__0

I tried solving it by breaking it up into two integrals:

x________________x
∫sqrt(5 + t^3) dt - ∫e^t^2 dt
0________________0

Then I tried using integration by parts, but that gets me things like du / 3t² = dt, and dv / 2t = dt. What do I do?

 

[LCKurtz]

Doing corrections on a test and I'm trying to solve this integral and I'm having quite a bit of trouble with it:

__x
D_x∫sqrt(5 + t^3) - e^t^2 dt
__0

I tried solving it by breaking it up into two integrals:

x________________x
∫sqrt(5 + t^3) dt - ∫e^t^2 dt
0________________0

Then I tried using integration by parts, but that gets me things like du / 3t² = dt, and dv / 2t = dt. What do I do?

You don't want to do the antidifferentiation. In fact, you can't, which is why you are having trouble. Differentiate the integral as a function of its upper limit using Leibnitz rule.

 

[STEMucator]

Is the Dx in front the derivative operator? If so you need to use the fundamental theorem of calculus to solve this.

 

[johnhuntsman]

You don't want to do the antidifferentiation. In fact, you can't, which is why you are having trouble. Differentiate the integral as a function of its upper limit using Leibnitz rule.

Im only in Calculus II and I've never heard of such a thing. I did some Googling and I definitely don't know what that is.

[Edit]Liebniz Rule isn't even mentioned in my textbook, and I'm almost certain it wasn't covered in class. Are you certain?[Edit]

 

[MarneMath]

Leibniz rule is just a generalization of the fundamental theorem of Calculus. It's useful to know, even if it isn't covered in your book. Anyway, as everyone has pointed out FToC is what you need here. Whenever you see one variable in the limit of integration and a different one within the integral, your mind should jump to this. Also if you ever see e^t^2, you should know that you cannot integrate this, and if you are being asked to solve it, there must be some 'trick' to it.

 

[johnhuntsman]

Alright then thanks. I guess I'ma watch some YouTube and go in for tutoring. No need to ask for a lesson here. I appreciate being pointed in the right direction : D Gracias.

 

[Ray Vickson]

Im only in Calculus II and I've never heard of such a thing. I did some Googling and I definitely don't know what that is.

[Edit]Liebniz Rule isn't even mentioned in my textbook, and I'm almost certain it wasn't covered in class. Are you certain?[Edit]

That is odd. When I looked up 'Leibnitz Rule' in Google I got hundreds of hits.

I would be willing to bet it WAS covered in class (although maybe not called that), because it is the very basis of doing integrals as "antiderivatives". It is absolutely fundamental, and any Calculus course worthy of the name will definitely cover it to some extent.

RGV

 

[SammyS]

I'm only in Calculus II and I've never heard of such a thing. I did some Googling and I definitely don't know what that is.

[Edit]Liebniz Rule isn't even mentioned in my textbook, and I'm almost certain it wasn't covered in class. Are you certain?[Edit]

I like to remember this as follows:


Let F(x) be the anti-derivative of f(x).

Then $\displaystyle \int_0^x f(t)\,dt=F(x)-F(0)\ .$

What is the derivative, with respect to x, of F(x) - F(0) ?

It's given by $\displaystyle \frac{d}{dx}(F(x)-F(0))=F'(x)-0=f(x)\ .$