question\ncalculate the integral and write the answer in simplest form.\n int (5 + 2x^{3}) dx

Explanation:

Step1: Apply sum - rule of integration

The sum - rule states that $\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$. So, $\int(5 + 2x^{3})dx=\int5dx+\int2x^{3}dx$.

Step2: Integrate each term

For $\int5dx$, since $\int kdx=kx + C$ (where $k = 5$), we have $\int5dx=5x+C_1$. For $\int2x^{3}dx$, using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), with $n = 3$ and a coefficient of 2, we get $\int2x^{3}dx=2\times\frac{x^{3+1}}{3 + 1}+C_2=\frac{2x^{4}}{4}+C_2=\frac{1}{2}x^{4}+C_2$.

Step3: Combine the results

Combining the two results, $\int(5 + 2x^{3})dx=5x+\frac{1}{2}x^{4}+C$, where $C = C_1+C_2$.

Answer:

$5x+\frac{1}{2}x^{4}+C$