determine the following ∫(3x²/√(x³ + 1))dx

Explanation:

Step1: Use substitution

Let $u = x^{3}+1$, then $du=3x^{2}dx$.

Step2: Rewrite the integral

The integral $\int\frac{3x^{2}}{\sqrt{x^{3}+1}}dx$ becomes $\int\frac{du}{\sqrt{u}}$.

Step3: Integrate

We know that $\int u^{n}du=\frac{u^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int\frac{du}{\sqrt{u}}=\int u^{-\frac{1}{2}}du$. Using the power - rule, we get $\frac{u^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+C = 2u^{\frac{1}{2}}+C$.

Step4: Substitute back

Substitute $u = x^{3}+1$ back into the result, we have $2\sqrt{x^{3}+1}+C$.

Answer:

$2\sqrt{x^{3}+1}+C$