`int_1^2((v^3 + 3v^6)/(v^4))dv` Evaluate the integral.

`int_1^2((v^3+3v^6)/v^4)dv`

simplify the integrand and apply the sum rule,

`=int_1^2(v^3/v^4+(3v^6)/v^4)dv`

`=int_1^2(1/v+3v^2)dv`

using the following common integrals

`int1/xdx=ln(x)` and `intx^n=(x^(n+1))/(n+1)` 

`=int_1^2(1/v)dv+3int_1^2v^2dv`

`=[ln(v)]_1^2+3[v^3/3]_1^2`

`=(ln(2)-ln(1)+3(2^3/3-1^3/3))`

`=ln(2)+3((8-1)/3)`

`=ln(2)+7`