The integral of e^x

1. May 18, 2012

robertjford80

1. The problem statement, all variables and given/known data

I thought sometimes the integral of e^x is xe^x. Under what circumstances is the integral of e^x = xe^x? I think it has something to do with u substitution.

2. May 18, 2012

Ray Vickson

Under NO circumstances is the integral of e^x equal to x e^x. I cannot imagine why you think that would hold.

RGV

3. May 18, 2012

robertjford80

4. May 18, 2012

sharks

The derivative of $e^x$ is: $$e^x .\frac{d(x)}{dx}$$

5. May 18, 2012

HallsofIvy

The derivative of $xe^x$ is, by the product rule $(x)'e^x+ (x)(e^x)'= 1(e^x)+ x(e^x)= xe^x+ e^x= (x+ 1)e^x$. As Ray Vickson said, the integral of $xe^x$ is NOT equal to itself and neither is the derivative.

The only functions having the property that their derivative is equal to the function itself is a constant times $e^x$.

6. May 18, 2012

sharks

I don't think anyone could have explained it better. This should resolve your confusion, robertjford80.

7. May 18, 2012

robertjford80

Here's an example

What's going on here? It clear says that the derivative of

c1e(3/2)x[ = (3/2)c1e(3/2)x

8. May 18, 2012

SammyS

Staff Emeritus
What's going on with the derivative of c1e(3/2)x[ is mainly the chain rule.

9. May 18, 2012

robertjford80

so the integral of e^2x is e^2x and the derivative of e^x is e^x but the derivative of e^2x is 2e^2x, is that right?

10. May 18, 2012

sharks

Correct, except for the integral of $e^{2x}$ which is $\frac{e^{2x}}{2}$

11. May 18, 2012

Number Nine

<deleted>

12. May 18, 2012

robertjford80

well, why don't you use the chain rule with e^x which would make it xe^x?

{this referred to number nine's deleted post} i saw it before he deleted it.

13. May 18, 2012

Villyer

The chain rule is to multiply by the derivative, and the derivative of x is 1.

If it helps, d/dx (ex) = 1 * ex

14. May 18, 2012

robertjford80

15. May 18, 2012

Ray Vickson

No, the first statement is not right, and is not what you asked originally. The indefinite integral of exp(a*x) for constant a is (1/a)*exp(a*x) + C; the derivative of exp(a*x) is a*exp(a*x). When a = 1 these both give just exp(x). For a = 2 they give (1/2) exp(2x) and 2 exp(2x), respectively.

RGV

16. May 18, 2012

sharks

That's exactly what i said before.

17. May 18, 2012

robertjford80

if you're talking about post 4, then i don't think you provided enough info to convey that

18. May 18, 2012

sharks

It's obvious that $\frac{dx}{dx}=1$ which gives $e^x .1=e^x$. Unless, you didn't know that, but it's really a basic notion of the principles of differentiation.
You should go over the basic principles, as it should help you to understand $e^x$ and the others more complicated that will follow.

19. May 18, 2012

robertjford80

if it was obvious i would not have posted the question

20. May 19, 2012

sankalpmittal

d(ex)/dx = ex

And so

∫ex dx = ___________ ....(i)

Note that integration is just reverse of differentiation.

If you want to verify this , then its simple ! Differentiate the left hand side of equation (i) with respect to x and see if its equal to ex. It will work.

If you want to prove it then analyze it by means of graph of f(x)=ex.

And note if you do this :

d(ex)/d(e) = xex-1

But it can never be xex !

Last edited: May 19, 2012