Lösung 2.3:1a

Aus Online Mathematik Brückenkurs 2

The formula for partial integration reads

fxgxdx=Fxgx−Fxgxdx ,

where Fx  is a primitive function of fx  and gx  is a derivative of gx .

If we are to use partial integration, the integrand has to be divided up into two factors, a factor fx  which we integrate and a factor gx  which we differentiate. It is only when the product Fxgx  becomes simpler than fxgx  that there is any point in partially integrating.

In the integral

2xe−xdx ,

it can seem appropriate to choose fx=e−x  and gx=2x , because then gx=2  and we have only Fx=−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:

=−2xe−x+2−e−x+C=−2xe−x−2e−x+C=−2x+1e−x+C