Solve Antiderivative of xe^x | Step-by-Step Guide

[dfklajsdfald]

Homework Statement

find the anti-derivative of x^e ^x so its x to the power of e to the power of x

Homework Equations

The Attempt at a Solution

i have 0 idea where to even start. this was a question on my quiz today

 

[BvU]

Hi dfk,

No homework. Still an attempt on your part required $$ F'(x) = x^{e^x} \ \Rightarrow \ F(x) = ??$$

 

[dfklajsdfald]

well i know that the derivative of e^x is e^x so the anti derivative of e^x is e^x and then my attempt would be e^xx but i don't know if that's right or even a good start

 

[dfklajsdfald]

Hi dfk,

No homework. Still an attempt on your part required $$ F'(x) = x^{e^x} \ \Rightarrow \ F(x) = ??$$

actually scratch the last part. my attempt would be x^{e^x +1} becacuse normally for any function x u go x and add one to the power and divide by it

 

[Mark44]

actually scratch the last part. my attempt would be x^{e^x +1} becacuse normally for any function x u go x and add one to the power and divide by it

So is $\frac d{dx}(x^{e^x + 1}) = x^{e^x}$?

For a power function, such as $x^n$, the derivative is $nx^{n - 1}$, but is that what you have here?

 

[epenguin]

Do you confirm that the formula in #2 is the one you meant?

Then you don't need to ask us whether a proposed solution is correct, surely you can differentiate?

If then you find it wrong, It may at least suggest something else to try. Or maybe some other direction to look.

 

[dfklajsdfald]

Do you confirm that the formula in #2 is the one you meant?

Then you don't need to ask us whether a proposed solution is correct, surely you can differentiate?

If then you find it wrong, It may at least suggest something else to try. Or maybe some other direction to look.

i don't know what ur asking, the question is find the antiderivative of the function x to the power of e to the power of x as stated at the start. I am not asking whether or not a solution is correct I am asking how abouts would i go to try and get the antiderivative because I've tried multiple ways (one which i have stated in a different reply) and can't seem to get there. I am only asking for guidance not for an answer of whether or not my solution is correct.

 

[dfklajsdfald]

So is $\frac d{dx}(x^{e^x + 1}) = x^{e^x}$?

For a power function, such as $x^n$, the derivative is $nx^{n - 1}$, but is that what you have here?

what i meant was that it was my attempt at finding the antiderivative of x^ex. i know that the derivative of e^x is e^x therefor the anti derivative would be the same. so then for derivatives with base x you go x^n-1 times the derivative of the power but since i need the anti derivative wouldn't i do xⁿ⁺¹ divided by the power instead? so wouldn't it be x^e^x +1 all divided by e^x(the plus one is added to e^x not to the entire function I am just having trouble showing it up on here properly)

 

[Mark44]

what i meant was that it was my attempt at finding the antiderivative of x^ex. i know that the derivative of e^x is e^x therefor the anti derivative would be the same. so then for derivatives with base x you go x^n-1 times the derivative of the power but since i need the anti derivative wouldn't i do xⁿ⁺¹ divided by the power instead?

That's not how it works -- you are applying the power rule for a function for which it doesn't apply.
Power rule: $\frac d{dx} x^n = nx^{n - 1}$
Here the exponent, n, is a constant. For your function the exponent, $e^x + 1$ is not a constant.

By your reasoning, $\frac d{dx} x^x =xx^{x - 1} = x^x$, which isn't true. The derivative of $x^x$ is actually $x^x(\ln x + 1)$.

so wouldn't it be x^e^x +1 all divided by e^x(the plus one is added to e^x not to the entire function I am just having trouble showing it up on here properly)

 

[Ray Vickson]

what i meant was that it was my attempt at finding the antiderivative of x^ex. i know that the derivative of e^x is e^x therefor the anti derivative would be the same. so then for derivatives with base x you go x^n-1 times the derivative of the power but since i need the anti derivative wouldn't i do xⁿ⁺¹ divided by the power instead? so wouldn't it be x^e^x +1 all divided by e^x(the plus one is added to e^x not to the entire function I am just having trouble showing it up on here properly)

To get the derivative, write $x^{e^x}= e^{f(x)}$, where $f(x) = \ln(x) e^x$, and use the chain rule.

 

[epenguin]

i don't know what ur asking, the question is find the antiderivative of the function x to the power of e to the power of x as stated at the start. I am not asking whether or not a solution is correct I am asking how abouts would i go to try and get the antiderivative because I've tried multiple ways (one which i have stated in a different reply) and can't seem to get there. I am only asking for guidance not for an answer of whether or not my solution is correct.

I thought and rereading #3 it still looks to me you were asking whether an answer was correct. We do often get to ask that question for integrals and have to point out that checking is the easy part - and also necessary. It seems to me you have now got some guidance to work on.

 

[dfklajsdfald]

To get the derivative, write $x^{e^x}= e^{f(x)}$, where $f(x) = \ln(x) e^x$, and use the chain rule.

ya but the question is asking for antiderivative

 

[Mark44]

ya but the question is asking for antiderivative

You claimed earlier that the antiderivative was $x^{e^x + 1}$. If this is correct, you should be able to differentiate this function to get your original integrand. That's what we're helping you with.

On the other hand, if you differentiate $x^{e^x + 1}$, and DON'T get $x^{e^x}$, then that tells you that your purported antiderivative is incorrect.

 

[dfklajsdfald]

You claimed earlier that the antiderivative was $x^{e^x + 1}$. If this is correct, you should be able to differentiate this function to get your original integrand. That's what we're helping you with.

On the other hand, if you differentiate $x^{e^x + 1}$, and DON'T get $x^{e^x}$, then that tells you that your purported antiderivative is incorrect.

ohh okay that makes more sense. and i wasnt claiming that that's what the antiderivative was. i was saying that was how i was trying to solve the antiderviative but was getting the wrong answer so i needed guidance to go abouts finding the antiderivative correctly. you asked what my initial attempt was which is why i responded with that, i know its wrong but u wanted to know what i did so that's why i showed that

 

[Mark44]

ohh okay that makes more sense. and i wasnt claiming that that's what the antiderivative was.

In post #4 you said:

actually scratch the last part. my attempt would be x^{e^x +1} becacuse normally for any function x u go x and add one to the power and divide by it

It looks to me like you were claiming that $x^{e^x + 1}$ is the antiderivative of $x^{e^x}$.

i was saying that was how i was trying to solve the antiderviative but was getting the wrong answer so i needed guidance to go abouts finding the antiderivative correctly. you asked what my initial attempt was which is why i responded with that, i know its wrong but u wanted to know what i did so that's why i showed that

Please don't use "text speak" here at PF. In the rules for this site (https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/ ), under General Posting Guidelines, it says"

SMS messaging shorthand ("text-message-speak"), such as using "u" for "you", "please" for "please", or "wanna" for "want to" is not acceptable.

 

[dfklajsdfald]

In post #4 you said:

It looks to me like you were claiming that $x^{e^x + 1}$ is the antiderivative of $x^{e^x}$.
Please don't use "text speak" here at PF. In the rules for this site (https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/ ), under General Posting Guidelines, it says"

i said my "attempt" i never claimed anything?? but anyway nevermind i'll figure it out on my own i just wanted some help and guidance not an argument or a forum of whose wrong whose right i'll delete this shortly