Question

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Explanation:

Step1: Simplify the integrand

Rewrite \(x\sqrt{3x}\) as \(\sqrt{3}x^{\frac{3}{2}}\). So the integral becomes \(\sqrt{3}\int x^{\frac{3}{2}}dx\).

Step2: Apply the power rule for integration

The power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). Here \(n=\frac{3}{2}\), then \(\sqrt{3}\times\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+C=\sqrt{3}\times\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+C\).

Step3: Simplify the expression

\(\sqrt{3}\times\frac{2}{5}x^{\frac{5}{2}}+C=\frac{2\sqrt{3}}{5}x^{\frac{5}{2}}+C\).

Answer:

\(\frac{2\sqrt{3}}{5}x^{\frac{5}{2}}+C\)