Select all expressions matching the value of indefinite integral int x^2(1-2x+x^2)dx (x^3(6x^2-15x+10))/(30)+C (x^3)/(3)(-x^2+(x^3)/(3))+C -(x^4)/(2)+(x^5)/(5)+C (x^3)/(3)-(x^4)/(2)+(x^5)/(5)+C

To find the indefinite integral of the given expression, we need to integrate the function \(x^2(1-2x+x^2)\) with respect to \(x\).<br /><br />Let's break down the expression inside the integral:<br /><br />\[ x^2(1-2x+x^2) = x^2 - 2x^3 + x^4 \]<br /><br />Now, we integrate each term separately:<br /><br />\[ \int (x^2 - 2x^3 + x^4) \, dx = \int x^2 \, dx - 2 \int x^3 \, dx + \int x^4 \, dx \]<br /><br />Integrating each term:<br /><br />\[ \int x^2 \, dx = \frac{x^3}{3} \]<br />\[ -2 \int x^3 \, dx = -2 \cdot \frac{x^4}{4} = -\frac{x^4}{2} \]<br />\[ \int x^4 \, dx = \frac{x^5}{5} \]<br /><br />Combining these results, we get:<br /><br />\[ \int (x^2 - 2x^3 + x^4) \, dx = \frac{x^3}{3} - \frac{x^4}{2} + \frac{x^5}{5} + C \]<br /><br />Therefore, the correct answer is:<br /><br />\[ \frac{x^3}{3} - \frac{x^4}{2} + \frac{x^5}{5} + C \]<br /><br />So, the correct option is:<br /><br />\[ \frac{x^3}{3} - \frac{x^4}{2} + \frac{x^5}{5} + C \]