Notes: Find y if dy/dx=xe^x?

Thursday, April 25, 2013

To determine the primitive function Y, we'll have to
calculate the indefinite integral of the given function:

x* $\displaystyle{{e}}^{{{x}}}$ dx = $\displaystyle\int$ dy = Y

We'll integrate by parts using
the formula:

udv = uv -
vdu

Let u = x => du =
dx

Let dv = $\displaystyle{{e}}^{{{x}}}$ dx => v =
$\displaystyle{{e}}^{{{x}}}$

x* $\displaystyle{{e}}^{{{x}}}$ dx = x* $\displaystyle{{e}}^{{{x}}}$ - $\displaystyle\int$ $\displaystyle{{e}}^{{{x}}}$
dx

x* $\displaystyle{{e}}^{{{x}}}$ = x* $\displaystyle{{e}}^{{{x}}}$ - $\displaystyle{{e}}^{{{x}}}$ +
C

The requested primitive function
is Y = (x-1)* $\displaystyle{{e}}^{{{x}}}$ + C.

Notes