<s>[INST] Solving the integral I'm trying to calculate the following integral :$\int \sqrt x (2-3x^2)^2dx$, but somehow I can't get it right, here's what I did:
First I expanded the expression:
\begin{align}&\int \sqrt x (4-12x^2 + 9x^4)
\\&= \int(4\sqrt x-12x^2\sqrt x + 9x^4\sqrt x) \end{align}
Then I evaluated each term individually:
\begin{align}&4 \int \sqrt x - 12 \int x ^\frac{5}{2} + 9 \int x^\frac{9}{2} \\&
= \frac{4x^\frac{3}{2} \cdot 2}{3} - \frac{12x^\frac{7}{2} \cdot 2}{7} + \frac{9x^\frac{11}{2} \cdot 2}{11}\end{align}
Which gives
$$\frac{8x^\frac{3}{2}}{3} - \frac{24x^\frac{7}{2}}{7} + \frac{18x^\frac{11}{2}}{11}$$
What's wrong with my solution?
 [/INST] You just forget the constant (and the minus sign as commented), other than that, your answer is correct.
We can verify the solution by differentiating the solution.
\begin{align}\frac{d}{dx}\left(\frac{8x^\frac{3}{2}}{3} - \frac{24x^\frac{7}{2}}{7} + \frac{18x^\frac{11}{2}}{11}+c\right) &=\left( \frac32\right)\frac{8x^\frac12}{3}-\left( \frac72\right)\frac{24x^\frac52}{7}+\left(\frac{11}2\right) \frac{18x^\frac92}{11}\\&= 4x^\frac12-12x^\frac52+9x^\frac92\\&= \sqrt x (4-12x^2 + 9x^4) \end{align}
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