Question

Calculați:
a. integrala de la 0 la 1 din
${xe}^{x}dx$
b. integrala de la 0 la 1 din
${xe}^{ {x}^{2} }dx$

Answer 1

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Answer 2

$\int_{0}^1xe^xdx = \int_{0}^1x\cdot(e^x)'dx = xe^x\Big|_0^1 - \int_{0}^1x'\cdot e^xdx = \\ \\ = 1\cdot e^1 - 0\cdot e^0 - \int_0^1e^xdx = e - (e^x)\Big|_0^1 = \\ \\ = e - (e^1-e^0) = e - (e-1) = 1 \\ \\ \\ \int_0^1xe^{x^2}dx = \int_0^1\frac{1}{2}\cdot 2 xe^{x^2}dx = \int_0^1\frac{1}{2}\cdot(x^2)'\cdot e^{x^2}dx = \\ \\ = \frac{1}{2}\cdot \int_0^1(x^2)'\cdot e^{x^2}dx = \frac{1}{2}\cdot \Big(e^{x^2}\Big)\Big|_0^1 = \\ \\ = \frac{1}{2}\cdot \Big(e^{1^2}-e^{0^2}\Big) = \frac{1}{2}\cdot (e-1) =$

$=\dfrac{e-1}{2}$