Найти первообразную [tex]\int{sin(x)e^x}[/tex]. Created by polka125. matematika-ru

то есть, мы получили:откуда

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July 2022 1 24 Report

Найти первообразную [tex]\int{sin(x)e^x}[/tex]

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spasibo3pajbrh
$\int{ \sin(x)e^x}dx = \\ = \int{ \sin(x)}d( {e}^{x} ) = \\ = {e}^{x} \sin(x) - \int{ {e}^{x} d(\cos(x) )} = \\ = {e}^{x} \sin(x) - ( {e}^{x} \cos(x) - \int( - \sin(x) ) {e}^{x} dx) = \\ = {e}^{x} ( \sin(x) - \cos(x) ) - \int{ \sin(x)e^x}dx \\$
то есть, мы получили:
$\int{ \sin(x)e^x}dx = \\ = {e}^{x} ( \sin(x) - \cos(x) ) - \int{ \sin(x)e^x}dx \\ \\$
откуда

$2\int{ \sin(x)e^x}dx = {e}^{x} ( \sin(x) - \cos(x) ) \\ \\ \int{ \sin(x)e^x}dx = \\ = \frac{1}{2} {e}^{x} ( \sin(x) - \cos(x) ) + c$

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