Integral of -e^(-x)

1. Sep 21, 2004

CinderBlockFist

Hi guys, I am looking in my calculus book, and it tells me in the example problem that the Integral of e^(-x)dx = -e^(-x)

I dont see how u get this answer. I know the integral of e^x = e^x but how does that negative sign come out to the left? Can someone explain this step by step, thanks.

2. Sep 21, 2004

recon

If you try differentiating -e-x, it comes to (-1)-e-x which is e-x. So it makes sense!

3. Sep 21, 2004

CinderBlockFist

Yes, but how do u get the integral without knowing thats the answer?

4. Sep 21, 2004

CinderBlockFist

Oh crap nevermind, its easy, just use u substitution u = -x

5. Sep 21, 2004

VinnyCee

This is a theorem.

The integral of e^u du = e^u * (du/dx).

6. Mar 22, 2009

roseSCC9

If you have e^(-x), you must divide e^(-x) by (-1), cuz (-1) is the coefficient of (-x)

that's the way you integrate

when you have e^(-3x) you must divide e^(-3x) by (-3)... so on...

7. Mar 23, 2009

Mentallic

or more simply, divide by the derivative of the power.

$$\int_a^b{e^{f(x)}}=\left [\frac{e^{f(x)}}{f'(x)} + C \right ]^b_a$$

8. Mar 23, 2009

ThirstyDog

I don't think that your last bit is true in general... I think that
$$\int_a^b{ f'(x) e^{f(x)}}dx=\left [e^{f(x)} + C \right ]^b_a$$
is what you mean and you have divided through. This is only correct when the f'(x) is constant... you might have known this, I just found it confusing the way you stated. Also for definite integrals you can omit the constant.

For example
$$\int_a^b{e^{-x^{2}}}=\left [\frac{\sqrt{\pi}}{2}erf(x) \right ]^b_a$$

9. Mar 23, 2009

HallsofIvy

That is completely wrong. In order to use "substitution" with u= f(x), f'(x), unless it is a constant, must already be in the integral.

10. Mar 23, 2009

slider142

You have been given:
$$\int e^u du = e^u + C$$
where C is an undetermined constant of integration.
You have come upon the integral:
$$\int f(g(x)) dx$$
where you know the indefinite integral of f(u), but not f(g(x)). In your case f(u) = eu and g(x) = -x. If we go back to derivatives, we may remember a related rule called the chain rule:
(f(g(x))' = f'(g(x))*g'(x)
In other words:
$$\int f'(g(x))*g'(x) dx = f(g(x)) + C$$
In some texts, they will use shorthand Leibnitz notation: Let u = g(x). Then du = g'(x) dx. The above integral is then written:
$$\int f'(u) du = f(u) + C$$
This is usually called u-substitution. Note that you have to have both f'(g(x)) = f'(u) and du = g'(x) dx in the integrand. Sometimes you will have to multiply the integrand by a creative version of 1 in order to make this happen. In your example, let f'(u) = eu since we already know how to integrate that and of course u = -x. Then du = -1 dx. Then we need our integrand to be f'(u) du = -e-x dx. Unfortunately, our integrand is actually -(e^-x dx) = -f'(u) du. Luckily, the -1 can be factored out of the integrand as it is a constant, so we have
$$-\int f'(u) du = - f(u) + C = -e^u + C$$

Last edited: Mar 23, 2009
11. Mar 23, 2009

deancodemo

Here is a simple rule you can use:
$$\int f(ax + b) \, dx = \frac{F(ax + b)}{a} + C$$

where F(x) is a primitive function of f(x). We can use this like so:
$$\int e^{-x} \, dx = \frac{e^{-x}}{-1} + C$$

12. Mar 24, 2009

Mentallic

Thanks for correcting me. It seems highschool doesn't expect any more than the elementary integrals of ef(x), f(x) being a linear function.
Hallsofivy, I'm not exactly sure what you're saying. This result has been given to us in class.

Maybe, $$\int f'(x)e^{f(x)}=\frac{f'(x)e^{f(x)}}{f'(x)}+C=e^{f(x)}+C$$

13. Mar 24, 2009

slider142

Halls is correct; your previous result makes no sense unless f(x) is linear, in which case f'(x) is just a constant, from which you can derive the result yourself from the chain rule. The far left and far right hand sides of the above quoted integral are correct, and is a straightforward result of the chain rule; your middle step is unjustified.

14. Jul 7, 2010

listen to this guy k.. the substitution method is sooooo not necessary in this case.. it just complicates something that is actually very very simple.. above is the general method of integrating exponentials.. it might be easier if u understand integration as the opposite of differentiation.. when differentiating exponentials you multiply by the derivative of function x.. and integrating is the opposite so you divide by the derivative of function x.. anyway good luck :)

15. Jul 7, 2010

Mentallic

Yes sorry, I was only thinking of functions f(x) that were linear y=ax+b.

16. Jul 7, 2010

Mentallic

And after reading the rest of the posts, it seems like my restriction has been mentioned multiple times.

The only quarrel they had was that the result I gave is only correct if f(x)=ax+b, $a\neq$0. It was my mistake. We should instead stick to the proper formula
And yes I agree with you. In this problem substitution should be avoided and that's what I try to get students to do - think of integrals as back-differentiation - when solving something so simple.

17. Sep 6, 2010

dazzwater

Question:

The integral of 2x.e^(x^2) is e^(x^2), isn't it?

Surely the integral of f'(x).e^f(x) is always e^f(x), regardless of whether f'(x) is a constant?

18. Sep 6, 2010

Mentallic

Yes, but what I did was rather than saying $$\int{f'(x)e^{f(x)}dx}=e^{f(x)}+c$$ I mistakenly wrote $$\int{e^{f(x)}dx}=\frac{e^{f(x)}}{f'(x)}+c$$ which is only true for f(x)=ax+b since then f'(x) is a constant.

19. Sep 6, 2010

Mu naught

u = -x
du = -1dx
dx = -du

so you subsitute e-x for eu and then substitute dx for -du, pull out the negative infront of the integral and you get -eu + C, subsituting back in and get:

-e-x + C

20. Sep 6, 2010

epenguin

A well kept secret - you practically can't get any integral without knowing that's the answer.

21. May 21, 2011

Naii

You've not done any mistake, you're right in both cases... it's 100% correct and if somebody is interested in fighting, i suggest them to visit university classes... It is correct and it's the same.

22. May 21, 2011

Mentallic

But it isn't, if you take the derivative of both sides, they won't be equal except in the special case that f'(x)=c for some constant $$c\neq 0$$

23. May 21, 2011

Naii

*sigh*
if you take it simple way, it is correct... if you want to digg in and know the answer for any e^f(x) ... you have a problem :) i've got here one problem i'm trying to figure out right now too, it's something like this:
$$\int{e^{\int{\frac{2x}{x^{2}+1}}dx}\int{4xe^{2x^{2}}dx}}dx$$
and hopefully it will show up correctly.. O.O it's quiet simple, i've got it done already but you can torture yourself too if you are interested hehe

24. May 21, 2011

Mentallic

Well isn't that what "100% correct" implies? :tongue:

Ahh well I haven't studied double integration yet but yeah if I were to take a stab at it, I would guess the answer is... 27

25. May 21, 2011

Naii

:D well... i'm just fighting this:
$$e^{-ln|x^{2}+1|}=\frac{e^{0}}{e^{ln|x^{2}+1|}}=\frac{1}{|x^{2}+1|}$$ any if this is correct? i'm not sure, that minus is uggly... whatever, i'll continue and see the result check if it is correct :)