Question

найдите значение ∫_0^ln2〖xe^x 〗dx=

Answer 1

$\int \limits ^{ \ln 2} _{0} x {e}^{x} dx = \begin{vmatrix}u = x; & dv = {e}^{x}dx \\ du = dx; & v = {e}^{x}\end{vmatrix} = \\ \\ = x {e}^{x} | ^{ \ln 2} _{0} - \int\limits ^{ \ln 2} _{0} {e}^{x} dx =( x {e}^{x} - {e}^{x} )| ^{ \ln 2} _{0} = \\ = \ln2 \times {e}^{ \ln2} - {e}^{\ln2} - 0 + {e}^{0} = \\=2\ln2 - 2 + 1 = 2\ln2 - 1$

Ответ: 2ln2 -1