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Aus Online Mathematik Brückenkurs 2

The formula for integration by parts reads

f(x)g(x)dx=F(x)g(x)−F(x)g(x)dx

where F(x) is a primitive function of f(x) and g(x) is a derivative of g(x).

If we are to use integration by parts, the integrand has to be divided up into two factors, a factor f(x) which we integrate and a factor g(x) which we differentiate. It is only when the product F(x)g(x) becomes simpler than f(x)g(x) that there is any point in integrating by parts.

In the integral

2xe−xdx

it can seem appropriate to choose f(x)=e−x and g(x)=2x, because then g(x)=2 and we have only F(x)=−e−x left,

2xe−xdx=2x−e−x−2−e−xdx=−2xe−x+2e−xdx.

It remains only to integrate e−x and we are finished,

\displaystyle \begin{align} \phantom{\int 2x\cdot e^{-x}\,dx}{} &= \rlap{-2xe^{-x} + 2\bigl(-e^{-x}\bigr) + C}\phantom{2x\cdot \bigl(-e^{-x}\bigr) - \int 2\cdot \bigl(-e^{-x}\bigr)\,dx}\\[5pt] &= -2xe^{-x} - 2e^{-x} + C\\[5pt] &= -2(x+1)e^{-x} + C\,\textrm{.} \end{align}