How does integration by parts work?

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How does integration by parts work?

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Answer 1 · 2014-09-12T18:40:05

Integration by Parts is like the product rule for integration, in fact, it is derived from the product rule for differentiation. It states
$\int u d v = u v - \int v d u$ $int u dv =uv-int v du$ .

Let us look at the integral
$\int x e^{x} d x$ $int xe^x dx$ .

Let $u = x$ $u=x$ .
By taking the derivative with respect to $x$ $x$
$\Rightarrow \frac{d u}{d x} = 1$ $Rightarrow {du}/{dx}=1$
by multiplying by $d x$ $dx$ ,
$\Rightarrow d u = d x$ $Rightarrow du=dx$

Let $d v = e^{x} d x$ $dv=e^xdx$ .
By dividing by $d x$ $dx$
$\Rightarrow \frac{d v}{d x} = e^{x}$ $Rightarrow {dv}/{dx}=e^x$
by integrating,
$\Rightarrow v = e^{x}$ $Rightarrow v=e^x$

Now, by Integration by Parts,
$\int x e^{x} d x = x e^{x} - \int e^{x} d x = x e^{x} - e^{x} + C$ $int xe^xdx =xe^x-inte^xdx=xe^x-e^x+C$

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How does integration by parts work?

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