The discussion focuses on the correct approach to solving the integral ∫(xe^x + e^x)dx using integration techniques. The initial attempt incorrectly applied integration by parts, while the correct method involves recognizing that the integral can be split into two separate integrals: ∫xe^x dx and ∫e^x dx. The latter can be solved directly, while the former requires integration by parts. The final solution is derived by combining the results of these two integrals.
Students and educators in calculus, particularly those focusing on integration techniques, as well as anyone looking to enhance their understanding of solving integrals involving exponential functions.
afcwestwarrior said:Homework Statement
∫e^x+e^x? Surely you don't mean [itex]2\int e^x dx[/itex]?
Are you now saying the problem is [itex]\int (xe^x+ e^x)dx[/itex]?Homework Equations
∫u dv= uv- ∫v du
The Attempt at a Solution
u= x+e^x
du= e^x
Then you are not using integration by parts, you are using a simple substitution. Yes, [itex]\int (xe^x+ e^x)dx= \int (x+ e^x)e^x dx[/itex]. If you let u= x+ ex, then du= (1+ e^x) dx, not just ex dx. And please by sure to include the "dx" in the integral; that may be part of what is confusing you.
Well, you can always check an integration yourself by differentiating.so it would be e^u
integral = e^u
= e^(e^x) +c is that correct, i know the answer is but what i just did
[tex]\frac{d}{dx}\left(e^{e^x}\right)= \frac{de^u}{du}\frac{de^x}{dx}[/tex]
with u= ex
[tex]= (e^u)(e^x)= (e^{e^x})(e^x)[/tex]
Which is not what you started with.
Did you consider just doing the two integrals separately?
[tex]\int (xe^x+ e^x)dx= \int xe^x dx+ \int e^x dx[/itex]<br /> You should be able to do the second of those directly and the first <b>is</b> a simple integration by parts.[/tex]