Question

When trying to evaluate the indefinite integral of (2x * e^x), why does using the substitution rule give me an incorrect answer?

(2x * e^x)
u = e^x
du/dx = e^x
dx = du/e^x

2 * integral of [ x*e^x du/e^x ]
e^x cancels out
2 * integral of [x]
answer is 2

Again ... why is this wrong?

Answer 1

because this is not a substitution method. you'll have to use integration by parts...

Answer 2

I know that much. I am wondering why the substitution method does not work in this case.

Answer 3

because as you can see, when you set u=e^x, there is still the factor of e^x when you calculate du... but the integrand only has an x (2x in this case)

Answer 4

Why can't I just cancel it out?

2 * integral of [x * e^x du/e^x]
It looks like it works (obviously it doesn't).

Answer 5

last step answer is not 2. 2 * integral of [x] is x^2

Answer 6

Put u=e^x as you mentioned, then du=e^x*dx .
And again everything should be in the function of U.
So,take u=e^x
Taking log on both sides,we get log(u)=log(e^x)
log(u)=xlog(e)
log(u)=x (loge=10
Coming back to the equation
integral of [2log(u)*u]
Using integral by parts we get the answer as xe^x-(e^x)/2

Answer 7

yes the do cancel but you'll then be left with integral x du. now since you're integrating wrt u, you'll need to change that x to lnu. so you have
integral lnu du.
even if you could do this new integral remember you need to switch back to the variabe x.