Lösung 2.3:1a

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,

=−2xe−x+2−e−x+C=−2xe−x−2e−x+C=−2(x+1)e−x+C.