+26387 Was ist die Funktion f(x)= (2x-1)*e^x aufgeleitet ?
$$\int{(2x-1)\cdot e^x} \ dx \\\\
=2\cdot \int{x\cdot e^x} \ dx - \int{ e^x} \ dx \quad | \quad \boxed{\int{ e^x} \ dx = e^x }\\\\
=2\cdot \int{x\cdot e^x} \ dx - e^x \\\\
\textcolor[rgb]{1,0,0}{
\small{\text{Wir l\"osen das $ \int{x\cdot e^x} \ dx $ mit partieller Integration }}
}\\\\\\
\small{\text{Wir wenden folgende Regel an : $
\boxed{\int{u\cdot v'} = u\cdot v - \int{u'\cdot v} } $}}\\\\
\small{\text{$\int{\underbrace{x}_{=u}\cdot \underbrace{e^x}_{=v'}} \ dx $}}\\ \\
\begin{array}{ll}
u = x & u' = 1 \\
v' = e^x & v = \int{v'} = \int{e^x}\ dx = e^x
\end{array}\\\\
\small{\text{$\int{\underbrace{x}_{=u}\cdot \underbrace{e^x}_{=v'}} \ dx = \underbrace{x}_{=u} \cdot \underbrace{e^x}_{=v} - \int{\underbrace{1}_{=u'}\cdot \underbrace{e^x}_{=v} }\ dx \\
$}}\\ \\
\small{\text{$\int{x \cdot e^x } \ dx = x \cdot e^x - \int{ e^x }\ dx
$}}\\ \\
\small{\text{$\int{x \cdot e^x } \ dx = x \cdot e^x - \underbrace{\int{ e^x }\ dx}_{=e^x}
$}}$$
$$\boxed{ \small{\text{$\int{x \cdot e^x } \ dx = x \cdot e^x - e^x
$}} }\\\\
\small{\text{
$
\int{(2x-1)\cdot e^x} \ dx \\
= 2\cdot \int{x\cdot e^x} \ dx - e^x
$}}\\
\small{\text{
$
\int{(2x-1)\cdot e^x} \ dx \\
= 2\cdot ( x \cdot e^x - e^x ) - e^x + c
$}}\\
\small{\text{
$\int{(2x-1)\cdot e^x} \ dx \\
= 2\cdot x \cdot e^x - 2 \cdot e^x - e^x + c
$}}\\
\small{\text{
$\int{(2x-1)\cdot e^x} \ dx \\
= 2\cdot x \cdot e^x - 3 \cdot e^x + c
$}} \\\\
\boxed{
\small{\text{
$\int{(2x-1)\cdot e^x} \ dx \\
= ( 2\cdot x - 3 ) \cdot e^x +c \
$}}
}$$
