daviddoji/CalculusWithJuliaNotes.jl
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use quarto, not Pluto to render pages

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CwJ/integrals/substitution.jmd
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@@ -247,8 +247,12 @@ But $u^3/3 - 4u/3 = (1/3) \cdot u(u-1)(u+2)$, so between $-2$ and $0$
it is positive and between $0$ and $1$ negative, so this integral is:
```math
\int_{-2}^0 (u^3/3 - 4u/3 ) du + \int_{0}^1 -(u^3/3 - 4u/3) du =
(\frac{u^4}{12} - \frac{4}{3}\frac{u^2}{2}) \big|_{-2}^0 - (\frac{u^4}{12} - \frac{4}{3}\frac{u^2}{2}) \big|_{0}^1 = \frac{4}{3} - -\frac{7}{12} = \frac{23}{12}.
\begin{align*}
\int_{-2}^0 (u^3/3 - 4u/3 ) du + \int_{0}^1 -(u^3/3 - 4u/3) du
&= (\frac{u^4}{12} - \frac{4}{3}\frac{u^2}{2}) \big|_{-2}^0 - (\frac{u^4}{12} - \frac{4}{3}\frac{u^2}{2}) \big|_{0}^1\\
&= \frac{4}{3} - -\frac{7}{12}\\
&= \frac{23}{12}.
\end{align*}
```
##### Example
@@ -400,13 +404,11 @@ Finally, we put back in the `u(x)` to get an antiderivative.
ex₃(w => u(x))
```
```julia; echo=false
note("""
Lest it be thought this is an issue with `SymPy`, but not other
systems, this example was [borrowed](http://faculty.uml.edu/jpropp/142/Integration.pdf) from an
illustration for helping Mathematica.
""")
```
!!! note
Lest it be thought this is an issue with `SymPy`, but not other
systems, this example was [borrowed](http://faculty.uml.edu/jpropp/142/Integration.pdf) from an
illustration for helping Mathematica.
## Trigonometric substitution
@@ -563,8 +565,8 @@ choices = [
"``\\int u (1 - u^2) du``",
"``\\int u \\cos(x) du``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -578,8 +580,8 @@ choices = [
"``u=\\sec(x)``",
"``u=\\sec(x)^2``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -593,8 +595,8 @@ choices = [
"``u=\\sqrt{x^2 - 1}``",
"``u=x``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -620,8 +622,8 @@ choices = [
"``\\int u du``",
"``\\int u^3/x du``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -633,8 +635,8 @@ choices = [
"``u=\\sin(x)``",
"``u=\\tan(x)``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -654,8 +656,8 @@ choices = [
"``a=0,~ b=0``",
"``a=1,~ b=1``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -669,8 +671,8 @@ choices = [
"``\\sec(u) = x``",
"``u = 1 - x^2``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -684,8 +686,8 @@ choices = [
"``\\tan(u) = x``",
"``\\sec(u) = x``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -699,8 +701,8 @@ choices = [
"``\\sec(u) = x``",
"``u = 1 - x^2``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
@@ -715,8 +717,8 @@ choices = [
"``\\sec(u) = x``",
"``4\\sin(u) = x``",
"``\\sin(u) = x``"]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -730,8 +732,8 @@ choices = [
"``\\tan(u) = x``",
"``a\\sec(u) = x``",
"``\\sec(u) = x``"]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -751,8 +753,8 @@ choices =[
"``a=\\pi/3,~ b=\\pi/2``",
"``a=1/2,~ b= 1``"
]
ans =1
radioq(choices, ans)
answ =1
radioq(choices, answ)
```
###### Question
@@ -763,6 +765,6 @@ How would we verify that $\log\lvert (\sec(u) + \tan(u))\rvert$ is an antideriva
choices = [
L"We could differentiate $\sec(u)$.",
L"We could differentiate $\log\lvert (\sec(u) + \tan(u))\rvert$ "]
ans = 2
radioq(choices, ans)
answ = 2
radioq(choices, answ)
```