Question

Please help me with my calculus homeworki am paying attention

Answer 1

`(x^3)/(3)+x+C`

1) In this integration, let's do it step by step

`\int (x^3+x)/(x)dx`

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:

`\begin{gathered} \int (x^3+x)/(x)dx \\ \int (x^3)/(x)+(x)/(x)dx \\ \int x^2+1dx \end{gathered}`

2.2) Now, we can start integrating it using the linearity and making use of the power rule for integration:

`\begin{gathered} \int x^2+1dx \\ (x^(2+1))/(2+1)+(x^(0+1))/(0+1)+C \\ (x^3)/(3)+x+C \end{gathered}`

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.