Lösung 2.1:2a

Aus Online Mathematik Brückenkurs 2

Version vom 10:15, 11. Mär. 2009

The foremost difficulty with calculating an integral is finding a primitive function of the integrand. Once we have done that, the integral is calculated as the difference between the primitive function's values in the upper and lower limits of integration.

The integrand in our case consists of two terms in the form xn, and so we can use the rule

xndx=xn+1n+1+C

on the terms individually to obtain that

F(x)=x2+12+1+3x3+13+1

is a primitive function of the integrand.

The integrand's value is thus

20x2+3x3dx= 3x3+34x420=323+3424−303+3404=38+4316=344.

Note: One way to check that F(x)=31x3+43x4 is a primitive function of the integral is to differentiate F(x) and to see that we obtain

F(x)=31x3+43x4=313x2+434x3=x2+3x3

as the integrand.