5.4 Indefinite Integrals and the Net Change Theorem/39 - Revision history

Dvaezazizi@laalliance.org: Protected "5.4 Indefinite Integrals and the Net Change Theorem/39" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2022-09-21T19:41:42Z

Protected "5.4 Indefinite Integrals and the Net Change Theorem/39" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 19:41, 21 September 2022
(No difference)

2022-09-21T16:29:24Z

← Older revision		Revision as of 16:29, 21 September 2022
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	&= \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(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]		&= \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] = \left[\frac{80}{5} + \frac{192}{5}\right] - \left[\frac{16}{5}\right]\\[2ex]

	&= \frac{256}{5}		&= \frac{256}{5}

2022-09-21T16:28:37Z

← Older revision		Revision as of 16:28, 21 September 2022
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	&= \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(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}		&= \frac{256}{5}

2022-09-21T16:24:36Z

← Older revision		Revision as of 16:24, 21 September 2022
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	&= \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[\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]

	~~&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{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(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]

2022-09-21T16:24:12Z

← Older revision		Revision as of 16:24, 21 September 2022
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	&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{1}^{64} \\[2ex]		&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{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]		&= \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]

			&= \frac{256}{5}

	\end{align}		\end{align}
	</math>		</math>


	~~= <math>2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})</math>~~

	~~= <math>16+38.4 - (2+1.2)</math>~~

	~~= <math>54.4 - 3.2</math>~~
	~~= <math>51.2</math>~~

	~~= <math>\frac{256}{5}</math>~~

2022-09-21T16:23:27Z

← Older revision		Revision as of 16:23, 21 September 2022
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	&= \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[\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]

	&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{1}^{64}		&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{1}^{64} \\[2ex]

	2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})		&= \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]

	\end{align}		\end{align}

2022-09-21T16:22:44Z

← Older revision		Revision as of 16:22, 21 September 2022
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	&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{1}^{64}		&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{1}^{64}

			2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})

	\end{align}		\end{align}
	</math>		</math>

	~~=<math>2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{1}^{64}</math>~~
	= <math>2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})</math>		= <math>2(64)^\frac{1}{2} + \frac{6}{5}(64)^\frac{5}{6} - (2(1)^\frac{1}{2} + \frac{6}{5}(1)^\frac{5}{6})</math>

2022-09-21T16:21:54Z

← Older revision		Revision as of 16:21, 21 September 2022
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	= \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]		= \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} = 2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6}		&= \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]

			&= 2(x)^\frac{1}{2} + \frac{6}{5}(x)^\frac{5}{6}\bigg\|_{1}^{64}

	\end{align}		\end{align}

2022-09-21T16:20:37Z

← Older revision		Revision as of 16:20, 21 September 2022
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	= \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]		= \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]

	&= \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+ \frac{x^{\frac{5}{6}}}{\frac{5}{6}} = 2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6}		&= \left[\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+ \frac{x^{\frac{5}{6}}}{\frac{5}{6}}\right]_{1}^{64} = 2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6}

2022-09-21T16:19:54Z

← Older revision		Revision as of 16:19, 21 September 2022
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	= \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]		= \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]

	= ~~\int_{1}^{64}~~\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+ \frac{x^{\frac{5}{6}}}{\frac{5}{6}} = ~~\int_{1}^{64}~~2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6}		&= \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+ \frac{x^{\frac{5}{6}}}{\frac{5}{6}} = 2x^\frac{1}{2} + \frac{6}{5}x^\frac{5}{6}