Find the general indefinite integral
∫ v(v3+5)2dv
∫v(v^3+5)^2 dv
= ∫v(v^6 + 10v^3 + 25) dv
= ∫v^7 + 10v^4 + 25v
Now just integrate each term.
To find the general indefinite integral of ∫ v(v^3+5)^2 dv, we can start by expanding the square term: (v^3+5)^2 = (v^3+5)(v^3+5).
Next, we multiply using the FOIL method to obtain: (v^3+5)(v^3+5) = v^6 + 5v^3 + 5v^3 + 25.
Simplifying further, we have: v^6 + 10v^3 + 25.
Now, we integrate each term individually:
∫ v^6 dv + ∫ 10v^3 dv + ∫ 25 dv.
Using the power rule of integration, we have:
(v^7)/7 + (10v^4)/4 + 25v + C.
Hence, the general indefinite integral of ∫ v(v^3+5)^2 dv is (v^7)/7 + (10v^4)/4 + 25v + C, where C is the constant of integration.