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

Question

$$\int\frac{5x^3+2}{\sqrt{x^3+1}}\,\mathrm{d}x$$

Sad to say this has really stumped me and nothing I have tried has worked.

I used Wolfram Alpha to find that the answer is simply $2x\sqrt{x^3+1}$ but it says the method is unavailable and I have no idea which method to use. Hints would be appreciated.

Hint: Take a look at my answer at http://math.stackexchange.com/questions/358452/evaluate-int-0-1-left-x-5-x-4-x-2-right/498977#498977 — Barry Cipra, Sep 20 '13 at 01:10

score 12 · Accepted Answer · answered Sep 20 '13 at 01:18

12

Observe that $$\int \frac{5x^3+2}{\sqrt{x^3+1}}dx=\int \frac{x(5x^3+2)}{x\sqrt{x^3+1}}dx=\int \frac{5x^4+2x}{\sqrt{x^5+x^2}}dx$$ and make the $u$-substitution $u=x^5+x^2$.

answered Sep 20 '13 at 01:18

azarel

13,120

1

Thanks, that's so simple! How does one spot tricks like these? – user85798 Sep 20 '13 at 01:29
@Oliver That's the same trick for computing $\int \sec x$. – azarel Sep 20 '13 at 01:33
How do you figure out which factor to multiply the top & bottom by? – user85798 Sep 20 '13 at 01:54
1

@Oliver Essentially I wanted to do the $u$-substitution $u=x^3+2$ but both polynomials have the same degree. Multiplying for $x$ increases the degree by one on top and two on the bottom because of the square root. But I wasn't expecting that would solve it directly. – azarel Sep 20 '13 at 02:16
Simple and profound! Nice. :) +1 – Ahaan S. Rungta Nov 20 '13 at 16:23

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

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