Get the answer to "I have this integral to evaluate:∫ex(1−ex)(1+ex)10dx"

basura8w081

Answered question

2022-03-31

I have this integral to evaluate:

\int e^{x} (1 - e^{x}) {(1 + e^{x})}^{10} d x

Jared Kemp

Beginner2022-04-01Added 14 answers

u = 1 + e^{x}

, then

e^{x} = u - 1

, so

1 - e^{x} = 1 - (u - 1) = 2 - u

. Now you can easily multiply out. Note that the ‘bad’ factor could be any simple polynomial in

e^{x}

, and the technique would still work. For instance, if the integrand had been

e^{x} (3 e^{2 x} - 4 e^{x} + 5) {(1 + e^{x})}^{10}

, substituting

u = 1 + e^{x}

would turn it into

(3 u^{2} - 2 u + 4) u^{10}

, since

3 e^{2 x} - 4 e^{x} + 5 = 3 {(u - 1)}^{2} + 4 (u - 1) + 5

= 3 u^{2} - 2 u + 4

Do you have a similar question?

Recalculate according to your conditions!