Mr. V Wiki Math - User contributions [en]

5.5 The Substitution Rule/27

2022-09-06T21:42:16Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \text{Substitute }t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:41:13Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \text{Substitute}t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:40:46Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \text{Substitute}t\text=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:38:52Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \text{Substitute}t=\sqrt[3]{1+z^3}\text \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:31:50Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \mathbf{Substitute} t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:30:02Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

Substitute t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:29:53Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:27:53Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \{{pad|4.0em}} t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:27:30Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

{{pad|4.0em}}t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:27:07Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& {{pad|4.0em}}t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:25:37Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& [[Wikipedia:Substitute|]]t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:24:33Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute
t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:24:24Z

Osbeenb61996:

5.5 The Substitution Rule/27

2022-09-06T21:23:58Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& SUBSTITUTE t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:23:09Z

Osbeenb61996:

5.5 The Substitution Rule/27

2022-09-06T21:22:55Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute [t]=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:21:47Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute {t}=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:16:47Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& [Substitute] t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:16:11Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute {t=\sqrt[3]{1+z^3}} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:15:53Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& {Substitute} t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:15:00Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute\\t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:14:50Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute\t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:14:41Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute\\\\t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:14:26Z

Osbeenb61996:

5.5 The Substitution Rule/27

2022-09-06T21:13:44Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& <poem> Substitute back t=\sqrt[3]{1+z^3} </poem> \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:12:30Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute back t=\sqrt[3]{1+z^3} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:12:17Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& Substitute back t=\sqrt[3]{1+z^3}

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:10:45Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} ______Substitute_back__ t=\sqrt[3]{1+z^3}\\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:10:26Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} Substitute back t=\sqrt[3]{1+z^3}\\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:08:05Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

&= \cfrac\sqrt[3]{(1+z^3)^2}{2} + C

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:07:50Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \cfrac\sqrt[3]{(1+z^3)^2}{2} + c

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:07:13Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \cfrac\sqrt[3]{(1+z^3)^2}{2}

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-06T21:07:03Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \cfrac{\sqrt[3]{(1+z^3)^2}{2}}

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-02T18:53:26Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:15:05Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \cfrac{\sqrt[3](1+z^3)^2{2} +c

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:14:37Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \cfrac{\sqrt[3]{(1+z^3)^2}{2} +c

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:13:30Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt} \\[2ex]

& \cfrac{t^2}{2} \\[2ex]

& \cfrac{\sqrt[3]{(1+z^3)^2}}{2} +c

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:11:45Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt}

& \cfrac{t^2}{2}

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:10:51Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz \\[2ex]

& \int{tdt}

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:09:55Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} dz

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:09:34Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\sqrt[3]{1+z^3}} +c

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:08:48Z

Osbeenb61996:

<math>
\begin{align}

& \int \cfrac{z^2}{\3sqrt{1+z^3}} +c

\end{align}
</math>

5.5 The Substitution Rule/27

2022-09-01T20:08:23Z

Osbeenb61996: Created page with "<math> \begin{align} & \int \cfrac{z^2}{3\sqrt{1+z^3}} +c \end{align} </math>"

<math>
\begin{align}

& \int \cfrac{z^2}{3\sqrt{1+z^3}} +c

\end{align}
</math>

5.4 Indefinite Integrals and the Net Change Theorem/25

2022-09-01T18:44:02Z

Osbeenb61996:

<math>
\begin{align}

& \int_{-2}^{2}({3u+1})^2 du \\[2ex]

&= \int {3u^2+6u+1} {du} \\[2ex]

&= {3u^3+3u^2+u}\bigg|_{-2}^{2} \\[2ex]

&= {3\cdot 2^3 + \cdot 2^2 +2 - 3\cdot -2^3 + 3 \cdot-2^2 -2} \\[2ex]

& = {52}

\end{align}
</math>

5.5 The Substitution Rule/7

2022-09-01T06:40:12Z

Osbeenb61996:

<math>
\begin{align}

& \int{x\cdot sin{(x^2)}} dx \\[2ex]

& \int \cfrac{sin(t)}{2} dt \\[2ex]

& \cfrac{1}{2} \cdot \int{sin(t)} dt \\[2ex]

& \cfrac{1}{2} \cdot (cos(x^2)) \\[2ex]

&= -\cfrac{cos(x^2)}{2} +C

\end{align}
</math>

5.5 The Substitution Rule/7

2022-09-01T06:39:22Z

Osbeenb61996:

5.5 The Substitution Rule/7

2022-09-01T06:38:59Z

Osbeenb61996:

5.5 The Substitution Rule/7

2022-09-01T06:33:49Z

Osbeenb61996:

<math>
\begin{align}

& \int{x\cdot sin{(x^2)}} dx \\[2ex]

& \int \cfrac{sin(t)}{2} dt \\[2ex]

& \cfrac{1}{2} \cdot \int{sin(t)} dt \\[2ex]

& \cfrac{1}{2} \cdot (cos(x^2))

\end{align}
</math>

5.5 The Substitution Rule/7

2022-09-01T06:28:41Z

Osbeenb61996:

<math>
\begin{align}

& \int{x\cdot sin{(x^2)}} dx \\[2ex]

& \int \cfrac{sin(t)}{2} dt \\[2ex]

& \cfrac{1}{2} \cdot \int{sin(t)} dt

\end{align}
</math>

5.5 The Substitution Rule/7

2022-09-01T06:26:49Z

Osbeenb61996:

<math>
\begin{align}

& \int{x\cdot sin{(x^2)}} dx \\[2ex]

& \int \cfrac{sin(t)}{2} dt

\end{align}
</math>