We use the following formula for integration:
[tex]\int x^ndx=\frac{x^{n+1}}{n+1}+C[/tex]
We have the following integral:
[tex]\int(15x^2+\frac{\sqrt[3]{x^2}}{4})dx[/tex]
Separate into two integrals:
[tex]\int(15x^2+\frac{\sqrt[3]{x^2}}{4})dx=\int15x^2dx+\int\frac{\sqrt[3]{x^2}}{4}dx[/tex]
Calculate the first integral. Take the coefficient out of the integral:
[tex]\int15x^2dx=15\int x^2dx[/tex]
Apply the integration formula:
[tex]\int15x^2dx=15\frac{x^3}{3}+C=5x^3+C[/tex]
Calculate the second integral. Take the coefficient out of the integral:
[tex]\int\frac{\sqrt[3]{x^2}}{4}dx=\frac{1}{4}\int\sqrt[3]{x^2}dx[/tex]
Express the radical as a fractional exponent:
[tex]\frac{1}{4}\int\sqrt[3]{x^2}dx=\frac{1}{4}\int x^{2/3}dx[/tex]
Apply the integration formula:
[tex]\frac{1}{4}\cdot\frac{x^{5/3}}{5/3}+C=\frac{3}{20}\sqrt[3]{x^5}+C[/tex]
The total integral is:
[tex]\int(15x^2+\frac{\sqrt[3]{x^2}}{4})dx=5x^3+\frac{3}{20}\sqrt[3]{x^5}+C[/tex]
