Evaluate $\int x^3(x^2+7)\ dx$ I'm trying to find the indefinite integral of $$\int x^3(x^2+7)\ dx$$ and I've seem to have forgotten how to do it in this case. So if anyone can refresh my memory, I'd appreciate it.

Take the integral: x^3(x^2+7)dx
For the integrand x^3(x^2+7), substitute u=x^2 and d u =2x dx -> 1/2 integral (u(u+7)) du
for the integrand u(u+7), substitute s=u+7 and d s=du -> 1/2 integral (s^2-7s)ds
integrate the sum term by term and factor out constants: 1/2 integral (s^2 ds)-7/2 integral (s)ds
the integral of s^2 is s^3/3 -> (s^3/6)-(7/2) integral(s)ds
the integral of s is s^2/2 -> (s^3/6)-(7s^2/4)+constant
substitute back for s=u+7 -> 1/6(u+7)^3 -(7/4)(u+7)^2+constant
substitute back for u=x^2 -> 1/6(x^2+7)^3 -(7/4)(x^2+7)^2+constant
factor the answer a different way -> 1/12(x^2+7)^2(2x^2-7)+constant
which is equivalent for restricted x values to -> x^6/6+7x^4/4+constant

SO:
Integral(x^3(x^2+7))dx=((x^6)/6)+((7x^4)/4)+constant
