# Thread: What is the best way to integrate this integral ?

1. ## What is the best way to integrate this integral ?

Integral $\int_{}^{} (9x^2+6)/(3x^3+6x) dx$

I understand to use the chain rule but,

i get ln(3x^3+6x) but the 9x^2+6 has me bit confused....hmm

2. ## Re: What is the best way to integrate this integral ?

The numerator is the derivative of the denominator. Whenever the integrand is of the form $\displaystyle \frac{f'(x)}{f(x)}$ you should try the substitution $\displaystyle u=f(x)$.

3. ## Re: What is the best way to integrate this integral ?

Originally Posted by bee77
Integral $\int_{}^{} (9x^2+6)/(3x^3+6x) dx$
I understand to use the chain rule but,
Hi Bee-Bee, learn to use this site.

4. ## Re: What is the best way to integrate this integral ?

Originally Posted by Plato
Hi Bee-Bee, learn to use this site.
You inadvertently entered "(9x^2+6)/(3x^3+6x^2)" into WolframAlpha, and that was what was last sitting there when I visited it.

It's really "(9x^2+6)/(3x^3+6x)."

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Originally Posted by bee77
Integral $\int_{}^{} (9x^2+6)/(3x^3+6x) dx$

I understand to use the chain rule but,

i get ln(3x^3+6x) but the 9x^2+6 has me bit confused....hmm

One of the forms of the answer is $\displaystyle ln|3x^3 + 6x| + C$, or you may see it as $\displaystyle log|3x^3 + 6x| + C$.

And, there are still other ways of writing the answer.

Regardless, it's the logarithm with the base e.

The graph of the function has a vertical asymptote of x = 0.