calculate the integral and write the answer in simplest form\n\n∫(2x^5 + 1 - 5x^(-3)) dx

Explanation:

Step1: Apply sum - difference rule

$\int(2x^{5}+1 - 5x^{-3})dx=\int 2x^{5}dx+\int 1dx-\int 5x^{-3}dx$

Step2: Use constant - multiple rule

$=2\int x^{5}dx+\int 1dx - 5\int x^{-3}dx$

Step3: Apply power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$

$2\times\frac{x^{5 + 1}}{5+1}+x-5\times\frac{x^{-3 + 1}}{-3 + 1}+C$ $=\frac{2x^{6}}{6}+x-\frac{5x^{-2}}{-2}+C$

Step4: Simplify the expression

$=\frac{1}{3}x^{6}+x+\frac{5}{2x^{2}}+C$

Answer:

$\frac{1}{3}x^{6}+x+\frac{5}{2x^{2}}+C$