$\int \frac{5x^3+2}{\sqrt{x^3+1}}dx$

Question

How do I integrate $\int \frac{5x^3+2}{\sqrt{x^3+1}}dx$ ? I know that the result is $2x\sqrt{x^3+1}$, but I cannot think of a way to get to it.

Answer 1

This is a subtle problem with integration by parts.

We begin by noting that integrals with $\sqrt{x^3+1}$ ordinarily require elliptic functions. Our objective is to try to eliminate the integral.

Begin by decomposing the integrand thusly, separating a term that has a higher power ($+1/2$ instead of $-1/2$) of the radicand:

$\dfrac{5x^3+2}{\sqrt{x^3+1}}=\dfrac{5x^3+5}{\sqrt{x^3+1}}-\dfrac{3}{\sqrt{x^3+1}}$

$=5\sqrt{x^3+1}-\dfrac{3}{\sqrt{x^3+1}}.$

So

$\int\dfrac{(5x^3+2)dx}{\sqrt{x^3+1}}=5\int\sqrt{x^3+1}dx-3\int\dfrac{dx}{\sqrt{x^3+1}}.$

We now integrate the first term on the right by parts:

$\int\dfrac{(5x^3+2)dx}{\sqrt{x^3+1}}=5x\sqrt{x^3+1}-5\int x[d\sqrt{x^3+1}]-3\int\dfrac{dx}{\sqrt{x^3+1}}+C$

$\text{[Properly, an indefinite integration by parts should include the constant.}$

$\text{Doing so avoids a fallacy in certain cases.]}$

$=5x\sqrt{x^3+1}-(5/2)\int\dfrac{3x^3 dx}{\sqrt{x^3+1}}-3\int\dfrac{dx}{\sqrt{x^3+1}}+C$

$=5x\sqrt{x^3+1}-\int\dfrac{[(15/2)x^3+3] dx}{\sqrt{x^3+1}}+C$

And then

$\color{blue}{\int\dfrac{(5x^3+2)dx}{\sqrt{x^3+1}}}=5x\sqrt{x^3+1}-\dfrac32\color{blue}{\int\dfrac{(5x^3+2)dx}{\sqrt{x^3+1}}}+C,$

in which the integrals in blue are now identical. We may therefore combine them on the left side and solve algebraically. Note that since $C$ is arbitrary, we need not multiply it by $2/5$ in the final result, which matches that given in the question.