∫x^2(3+4x)^2 dx
x^2*(3+4x)^2=x^2*(9+24x+16x^2)=9x^2+24x^3+32x^4
∫ x^2*(3+4x)^2dx= ∫ (9x^2+24x^3+32x^4)dx=9 ∫ x^2dx+24 ∫ x^3dx+32 ∫ x^4dx=
=9*(x^3/3)+24*(x^4/4)+32*(x^5/5)+C=3*x^3+6*x^4+(32/5)*x^5+C
Проверка:
(3*x^3+6*x^4+(32/5)x^5+C)`=(3*x^3)`+6*(x^4)`+(32/5)*(x^5)`+(C)`=9x^2+24x^3+32x^4=x^2*(9+24x+32x^2)=x^2*(3+4x)^2