Question

calculate the integral and write your answer in simplest form. $int \frac{4sqrt3{x^5}}{3}dx$

Explanation:

Step1: Rewrite integrand with exponents

$\int \frac{4x^{\frac{5}{3}}}{3} dx = \frac{4}{3} \int x^{\frac{5}{3}} dx$

Step2: Apply power rule for integration

The power rule is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n
eq -1$. Here $n=\frac{5}{3}$, so:
$\frac{4}{3} \cdot \frac{x^{\frac{5}{3}+1}}{\frac{5}{3}+1} + C = \frac{4}{3} \cdot \frac{x^{\frac{8}{3}}}{\frac{8}{3}} + C$

Step3: Simplify the coefficients

$\frac{4}{3} \cdot \frac{3}{8} x^{\frac{8}{3}} + C = \frac{1}{2} x^{\frac{8}{3}} + C$

Step4: Rewrite in radical form (optional, simplest form)

$\frac{1}{2} \sqrt[3]{x^8} + C = \frac{1}{2} x^2 \sqrt[3]{x^2} + C$

Answer:

$\frac{1}{2}x^{\frac{8}{3}} + C$ (or $\frac{1}{2}x^2\sqrt[3]{x^2} + C$)