[tex]\frac{x^3}{3}+x+C[/tex]
1) In this integration, let's do it step by step
[tex]\int \frac{x^3+x}{x}dx[/tex]
Before integrating it, let's rewrite it so that our job can become more convenient.
2) So, we can rewrite it like that, and simplify it before integrating it:
[tex]\begin{gathered} \int \frac{x^3+x}{x}dx \\ \int \frac{x^3}{x}+\frac{x}{x}dx \\ \int x^2+1dx \end{gathered}[/tex]
2.2) Now, we can start integrating it using the linearity and making use of the power rule for integration:
[tex]\begin{gathered} \int x^2+1dx \\ \frac{x^{2+1}}{2+1}+\frac{x^{0+1}}{0+1}+C \\ \frac{x^3}{3}+x+C \end{gathered}[/tex]
Notice that this is not defined so we can write the Constant in a general way: C. Notice also that x^0 = 1 so that's why we substituted 1 by x^0
And that's the answer.
