The integral \(\int_0^1 \sqrt{x} - x^3 \, dx\) can be evaluated using the power rule for integration. The expression simplifies to \(\int_0^1 x^{1/2} \, dx - \int_0^1 x^3 \, dx\). Applying the power rule, the integral of \(x^{1/2}\) results in \(\frac{2}{3} x^{3/2}\) evaluated from 0 to 1, yielding \(\frac{2}{3}\). The integral of \(x^3\) results in \(\frac{1}{4} x^4\) evaluated from 0 to 1, yielding \(\frac{1}{4}\). Thus, the final result is \(\frac{2}{3} - \frac{1}{4} = \frac{5}{12}\).
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