Integrate 2x/4  3x  x^2 dx
Understand the Problem
The question is asking to perform the integration of the polynomial expression (2x/4  3x  x^2) with respect to x. We will apply the rules of calculus to find the integral of this expression.
Answer
The integral is $ \frac{5}{4}x^2  \frac{x^3}{3} + C $.
Answer for screen readers
The integral of the expression $\frac{2x}{4}  3x  x^2$ with respect to $x$ is: $$ \frac{5}{4}x^2  \frac{x^3}{3} + C $$
Steps to Solve

Rewrite the expression First, rewrite the polynomial expression $ \frac{2x}{4}  3x  x^2 $ in a simpler form for integration: $$ \frac{2x}{4} = \frac{1}{2}x $$ So the expression becomes: $$ \frac{1}{2}x  3x  x^2 $$

Combine like terms Combine the terms with $x$: $$ \frac{1}{2}x  3x = \left(\frac{1}{2}  3\right)x = \frac{5}{2}x $$ Therefore, the expression simplifies to: $$ \frac{5}{2}x  x^2 $$

Set up the integral Now set up the integral of the expression: $$ \int \left(\frac{5}{2}x  x^2 \right) , dx $$

Integrate each term Integrate each term separately:
 For $\frac{5}{2}x$, its integral is: $$ \int \frac{5}{2}x , dx = \frac{5}{2} \cdot \frac{x^2}{2} = \frac{5}{4}x^2 $$
 For $x^2$, its integral is: $$ \int x^2 , dx = \frac{x^3}{3} $$
 Combine the results Combine the results of the integrals: $$ \frac{5}{4}x^2  \frac{x^3}{3} + C $$ where $C$ is the constant of integration.
The integral of the expression $\frac{2x}{4}  3x  x^2$ with respect to $x$ is: $$ \frac{5}{4}x^2  \frac{x^3}{3} + C $$
More Information
This integral represents the area under the curve of the polynomial expression between two points. Integration is a fundamental concept in calculus that allows us to determine the accumulated value of a function over an interval.
Tips
 Forgetting to combine like terms before integrating can lead to incorrect results.
 Not including the constant of integration $C$ at the end of the integral.