Key to "Evaluate the following definite integrals ∫01xe(−x2+2)dx"

SchachtN

Answered question

2021-02-18

Evaluate the following definite integrals

\int_{0}^{1} x e^{(- x^{2} + 2)} d x

i1ziZ

Skilled2021-02-19Added 92 answers

Step 1
To Evaluate the following definite integrals:
Step 2
Given that

\int_{0}^{1} x e^{(- x^{2} + 2)} d x

Let

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

\frac{d u}{d x} = - 2 x (e^{- x^{2} + 2})

\frac{d u}{- 2} = x e^{(- x^{2} + 2)} d x

Hence

\int_{0}^{1} x e^{(- x^{- 2} + 2)} d x = \int_{0}^{1} \frac{d u}{- 2}

\frac{- 1}{2} {[4]}_{0}^{1}

= \frac{- 1}{2} [e^{1} - e^{2}]

= \frac{- 1}{2} [e - e^{2}] = \frac{1}{2} [e^{2} - e]

∵ \int_{0}^{1} x e^{(- x^{2} + 2)} d x = \frac{1}{2} (e^{2} - e)

Jeffrey Jordon

Expert2021-11-05Added 2605 answers

Answer is given below (on video)

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