integral(-6sqrtx)/(1+x^3) do the whole thing

We are tasked with evaluating the integral:

\[
\int \frac{-6\sqrt{x}}{1 + x^3} \, dx
\]

### Step 1: Substitution to simplify the integrand
Let’s first introduce a substitution that might simplify the problem. Since we have a term involving \(x^3\) in the denominator and a square root of \(x\) in the numerator, we can use a substitution that could reduce these complexities.

Let:
\[
u = x^{3/2}
\]
This gives us:
\[
du = \frac{3}{2} x^{1/2} dx \quad \text{or equivalently} \quad dx = \frac{2}{3} \frac{du}{x^{1/2}}
\]

Now, since \(x^{1/2} = \sqrt{x}\), we rewrite the integral as:

\[
\int \frac{-6 \sqrt{x}}{1 + x^3} \, dx
\]

Substituting \(dx = \frac{2}{3} \frac{du}{x^{1/2}}\) and \(x^{3/2} = u\),

Learn how to evaluate the integral of (-6sqrt(x))/(1 + x^3) with a step-by-step solution. This problem uses substitution to simplify the integrand and involves key concepts in calculus such as substitution and algebraic manipulation. Suitable for college-level students studying calculus.

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