determine the following: ∫5x^(3/2)dx

Explanation:

Step1: Use power - rule for integration

The power - rule for integration is $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For the integral $\int 5x^{\frac{3}{2}}dx$, we can take out the constant factor 5 first. So $\int 5x^{\frac{3}{2}}dx = 5\int x^{\frac{3}{2}}dx$.

Step2: Apply the power - rule formula

Here $n=\frac{3}{2}$. Using the formula $\int x^n dx=\frac{x^{n + 1}}{n + 1}+C$, we have $5\times\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+C$.

Step3: Simplify the expression

First, calculate $\frac{3}{2}+1=\frac{3 + 2}{2}=\frac{5}{2}$. Then $5\times\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+C$. Since $5\times\frac{2}{5}x^{\frac{5}{2}}+C$, the simplified result is $2x^{\frac{5}{2}}+C$.

Answer:

$2x^{\frac{5}{2}}+C$