Evaluate $\int x^3(x^2+7)\ dx$

Question

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.

Answer 1

First expand the integrand.

It simplifies to $x^5+7x^3$. Then we have that

$$\int x^5 + 7x^3=\frac{x^6}{6}+\frac{7x^4}{4}+C$$

This is done by the Power Rule, i.e, $$\int x^n\ dx=\frac{x^{n+1}}{n+1}+C$$

Answer 2

I think this problem simply appears more complicated than it is (i.e. you may be looking for a substitution, but one is not needed). Just multiply first: $x^5 + 7x^3$ is simple to integrate.

Answer 3

just open the bracket, and do each one separately, this should be easy one.

Answer 4

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

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SO:

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