Integrate $\frac{2x^3 + 2x}{x}$ with respect to x

Analysis

The question asks us to find the indefinite integral of the function $\frac{2x^3 + 2x}{x}$ with respect to $x$. Before we can integrate, we must simplify the algebraic expression to make it easier to handle.

Concept: Integration of Algebraic Functions

To integrate a polynomial, we use the Power Rule for Integration:

where $k$ is the constant of integration.

Step-by-Step Solution

Step 1: Simplify the expression Divide each term in the numerator by the denominator ($x$):

Step 2: Set up the integral Now, integrate the simplified expression with respect to $x$:

Step 3: Integrate term by term Apply the power rule to each term:

• For $2x^2$: $\int 2x^2 \, dx = 2 \left( \frac{x^{2+1}}{2+1} \right) = \frac{2x^3}{3}$
• For $2$: $\int 2 \, dx = 2x$

Step 4: Combine and add the constant

Conclusion

Comparing our result to the given options, we find that our solution matches option C.

Exam Tip

Whenever you see a fraction in an integration problem, always check if you can simplify it by dividing the numerator by the denominator first. It usually turns a complex-looking problem into a very simple one! Also, never forget to add the constant $k$ (or $c$) at the end of every indefinite integral; examiners often penalize for missing it.