<s>[INST] Question about writing integrals as a sum of integrals? So the question that is given is as follows:
Write the integral as a sum of integrals without absolute values and evaluate
$$ \int_{-2}^3 |x^3| \, dx$$
Can someone help me figure this out because I keep getting $20.25$ or $\dfrac{81}{4}$ but the answer is supposedly $24.25$ or $\dfrac{97}{4}$.
 [/INST] Note that $|x^3| = -x^3$ when $x \le 0$. We can therefore split the integral into two as follows:
$$\int_{-2}^3 |x^3| dx = \int_{-2}^0 |x^3| dx + \int_0^3 |x^3| dx = -\int_{-2}^0 x^3 dx + \int_0^3 x^3 dx = -\frac{x^4}{4}\bigg|_{-2}^0 + \frac{x^4}{4}\bigg|_0^3$$
$$= -\left(\frac{0}{4}-\frac{16}{4}\right) + \left(\frac{81}{4}-\frac{0}{4}\right) = \frac{16}{4}+\frac{81}{4} = \frac{97}{4}.$$
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