- #1
afcwestwarrior
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Homework Statement
∫e^x+e^x
Homework Equations
∫u dv= uv- ∫v du
The Attempt at a Solution
u= x+e^x
du= e^x
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
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]
You should be able to do the second of those directly and the first is a simple integration by parts.
The general formula for integration by parts is ∫u dv = uv - ∫v du, where u is the first function and dv is the second function. This formula is used to solve integrals where one function is difficult to integrate, but the product of two functions is easier to integrate.
The u-substitution rule can be used to determine which function to use as u. This rule states that the function that appears first when performing the derivative (from left to right) should be used as u. The remaining function will then be used as dv.
No, integration by parts can only be used on integrals where the product of two functions is present. It cannot be used on integrals that do not have this form.
Integration by parts can be used multiple times on a single integral if necessary. Each time it is used, the integral becomes simpler and more manageable.
Yes, there are certain rules that should be followed when using integration by parts. These include using the u-substitution rule to determine which function to use as u, choosing the correct dv function, and applying the formula correctly. It is also important to pay attention to any special cases, such as when the integral becomes an infinite series or when it becomes a repeated integral.