Lösung 2.2:1a

Aus Online Mathematik Brückenkurs 2

A substitution of variables is often carried out so as to transform a complicated integral to one that is less complicated which one can either directly calculate or continue to work with.

When we carry out a substitution of variables u=ux , there are three things which are affected in the integral:

1. the integral must be rewritten in terms of the new variable u; 2. the element of integration, dx, is replaced by du, according to the formula du=uxdx ; 3. the limits of integration are for x and must be changed to limits of integration for the variable u.

In this case, we will perform the change of variables u=3x−1, mainly because the integrand 13x−14 will then be replaced by 1u4.

The relation between dx and du reads

du=uxdx=3x−1dx=3dx ,

which means that dx is replaced by 31du.

Furthermore, when x=1 in the lower limit of integration, the corresponding u -value becomes u=31−1=2, and when x=2, we obtain the u-value u=32−1=5

One usually writes the whole substitution of variables as

21dx3x−14=u=3x−1du=3dx=52u431du 

Sometimes, we are more brief and hide the details:

21dx3x−14=u=3x−1=52u431du 

After the substitution of variables, we have a standard integral which is easy to compute.

In summary, the whole calculation is:

21dx3x−14=u=3x−1du=3dx=52u431du=3152u−4du=u−4+1−4+152=−911u352=−91153−123=−91235323−53=117322353=3213322353=132353=131000.