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

To determine which expressions match the value of the indefinite integral $\int x^{2}(1-2x+x^{2})dx$, we need to evaluate the integral.<br /><br />First, let's simplify the integrand:<br />$x^{2}(1-2x+x^{2}) = x^{2} - 2x^{3} + x^{4}$<br /><br />Now, we can integrate term by term:<br />$\int x^{2} dx - \int 2x^{3} dx + \int x^{4} dx$<br /><br />Integrating each term separately:<br />$\int x^{2} dx = \frac{x^{3}}{3}$<br />$\int 2x^{3} dx = x^{4}$<br />$\int x^{4} dx = \frac{x^{5}}{5}$<br /><br />Putting it all together:<br />$\int x^{2}(1-2x+x^{2})dx = \frac{x^{3}}{3} - x^{4} + \frac{x^{5}}{5} + C$<br /><br />Now, let's compare this result with the given options:<br /><br />1. $\frac{x^{3}(6x^{2}-15x+10)}{30}+C$:<br /> Simplifying this expression:<br /> $\frac{x^{3}(6x^{2}-15x+10)}{30}+C = \frac{1}{30}(6x^{5}-15x^{4}+10x^{3})+C$<br /> $= \frac{1}{5}x^{5} - \frac{1}{2}x^{4} + \frac{1}{3}x^{3} + C$<br /> This expression does not match the result we obtained.<br /><br />2. $\frac{x^{3}}{3}(-x^{2}+\frac{x^{3}}{3})+C$:<br /> Simplifying this expression:<br /> $\frac{x^{3}}{3}(-x^{2}+\frac{x^{3}}{3})+C = -\frac{x^{5}}{3} + \frac{x^{6}}{9} + C$<br /> This expression does not match the result we obtained.<br /><br />3. $-\frac{x^{4}}{2}+\frac{x^{5}}{5}+C$:<br /> This expression matches the result we obtained.<br /><br />Therefore, the correct answer is:<br />$-\frac{x^{4}}{2}+\frac{x^{5}}{5}+C$