find the most general antiderivative or indefinite integral. ∫3x^√3 dx ∫3x^√3 dx = □

Explanation:

Step1: Use constant - multiple rule

The constant - multiple rule of integration states that $\int cf(x)dx = c\int f(x)dx$, where $c$ is a constant. Here $c = 3$ and $f(x)=x^{\sqrt{3}}$. So, $\int 3x^{\sqrt{3}}dx=3\int x^{\sqrt{3}}dx$.

Step2: Apply power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$, where $n\neq - 1$ and $C$ is the constant of integration. For $n=\sqrt{3}$, we have $\int x^{\sqrt{3}}dx=\frac{x^{\sqrt{3}+1}}{\sqrt{3}+1}+C$.

Step3: Multiply by the constant

$3\int x^{\sqrt{3}}dx = 3\times\frac{x^{\sqrt{3}+1}}{\sqrt{3}+1}+C=\frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1}+C$.

Answer:

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