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
$int u dv =uv-int v du$ .

Let us look at the integral
$int xe^x dx$ .

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

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

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

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How does integration by parts work?

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