5.4 Indefinite Integrals and the Net Change Theorem/39

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{1}^{64}\frac{1+\sqrt[3]{x}}\sqrt{x}dx &= \int_{1}^{64}\left(\frac{1}{x^{1/2}} + \frac{x^{1/3}}{x^{1/2}}\right)dx = \int_{1}^{64}\left(x^{-1/2}+x^{\frac{1}{3}-{\frac{1}{2}}}\right)dx = \int_{1}^{64}\left(x^{-\frac{1}{2}}+x^{-\frac{1}{6}}\right)dx \\[2ex] &= \left[\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+ \frac{x^{\frac{5}{6}}}{\frac{5}{6}}\right]_{1}^{64} = \left[2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6}\right]_{1}^{64} \\[2ex] &= \left[2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6}\right] - \left[(2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})\right] \\[2ex] &= \left[2\cdot8 + \frac{6}{5}(2)^5\right] - \left[2+\frac{6}{5}\right] = \left[16+\frac{192}{5}\right] - \left[\frac{16}{5}\right] \\[2ex] &= \frac{256}{5} \end{align} }