f(x)=x(2+x^2)^2 Answer: ((x^2+2)^3)/(6)+C

The provided answer is incorrect. Let's find the correct integral of $f(x) = x(2 + x^2)^2$.<br /><br />We can solve this using substitution. Let $u = 2 + x^2$. Then, the derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 2x$, which means $du = 2x \, dx$, or $x \, dx = \frac{1}{2} du$.<br /><br />Now, substitute $u$ and $du$ into the integral:<br /><br />$\int x(2 + x^2)^2 dx = \int u^2 \cdot \frac{1}{2} du = \frac{1}{2} \int u^2 du$<br /><br />Now, integrate with respect to $u$:<br /><br />$\frac{1}{2} \int u^2 du = \frac{1}{2} \cdot \frac{u^3}{3} + C = \frac{u^3}{6} + C$<br /><br />Finally, substitute back the original expression for $u$:<br /><br />$\frac{(2 + x^2)^3}{6} + C$<br /><br />Therefore, the correct answer is $\frac{(2 + x^2)^3}{6} + C$.<br />