Question

A integral $\displaystyle\int_1^e \int_1^{e^2}\int_1^{e^3}\left(\frac{1}{xyz}\right )dx\,dy\,dz$ vale:

a) 4

b) 2

c) 3

d) 1

e) 6

Answer 1

int (1,e) int (1,e^2)int (1,e^3) 1/xyz dxdydz
int (1,e) int (1,e^2) ln x/yz  |(1,e^3)  dydz
int (1,e) int (1,e^2) ln e^3/yz-ln e^1/yz   dydz
int (1,e) int (1,e^2) 3ln e/yz- 0/yz   dydz
int (1,e) int (1,e^2) 3 /yz dydz
3int (1,e) ln y/z | (1,e^2) dz
3int (1,e) ln e^2 /z -ln1/z) dz
3 int (1,e) 2ln e/z-0/z dz
3 int (1,e) 2/z dz
3 ln z | (1,e)
3.2 (ln e-ln 1) =3.2(1-0)=6