Find this definite integral ∫ 1 2 e 1 / x x 4 d x

Question

Find this definite integral      ∫          1              2                  e              1                  /                x                    x      4        d  x

Kaya Kemp · Accepted Answer

I think you made an algebra mistake when integrating by parts. Starting with

\int u^{2} e^{u} d u,

your choice of

g = u^{2}, d f = e^{u} d u

is good. This yeilds

\int u^{2} e^{u} d u = u^{2} e^{u} - \int 2 u e^{u} d u = u^{2} e^{u} - 2 \int u e^{u} d u .

Next, we focus on

\int u e^{u} d u

. Like you suggested, we pick

g = u, d f = e^{u} d u

. This gives us

\int u e^{u} d u = u e^{u} - \int e^{u} d u = u e^{u} - e^{u} + c .

Putting everything together, we have

\int u^{2} e^{u} d u = u^{2} e^{u} - 2 (u e^{u} - e^{u} + c) = e^{u} (u^{2} - 2 u + 2) + c .

I think you forgot to correctly distribute the factor of 2, and a minus sign.

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