Lösung 2.1:2a

Aus Online Mathematik Brückenkurs 2

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

Fx=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 Fx=31x3+43x4  is a primitive function of the integral is to differentiate Fx  and to see that we obtain

Fx=31x3+43x4=313x2+434x3=x2+3x3

as the integrand.