What is $f (x) = \int x e^{x}$ $f(x) = int xe^x$ if $f (2) = 3$ $f(2)=3$ ?

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Answer 1 · 2018-03-05T15:08:46

$f (x) = x e^{x} - e^{x} + 3 - e^{2}$ $f(x)=xe^x-e^x+3-e^2$

$f (x) = \int x e^{x} d x, f (2) = 3$ $f(x)=intxe^xdx, f(2)=3$

$f (x) = \int u \frac{d v}{d x} d x = u v - \int v \frac{d u}{d x} d x$ $f(x)=intu(dv)/(dx)dx=uv-intv(du)/(dx)dx$

in this case

$u = x \Rightarrow \frac{d u}{d x} = 1$ $u=x=>(du)/(dx)=1$

$\frac{d v}{d x} = e^{x} \Rightarrow v = e^{x}$ $(dv)/(dx)=e^x=>v=e^x$

$:.f(x)=xe^x-inte^xdx$

$f(x)=xe^x-e^x+c$

$f(2)=3$

$:. f(2)=3=2e^2-e^2+c$

$c=3-e^2$

$f(x)=xe^x-e^x+3-e^2$

What is f(x) = int xe^xf(x)=∫xexf(x) = int xe^x if f(2)=3 f(2)=3f(2)=3 ?