diff --git a/chapters/ch5.jl b/chapters/ch5.jl
index 0467e85..b49e8e5 100644
--- a/chapters/ch5.jl
+++ b/chapters/ch5.jl
@@ -1,5 +1,5 @@
### A Pluto.jl notebook ###
-# v0.19.19
+# v0.19.26
using Markdown
using InteractiveUtils
@@ -16,31 +16,31 @@ end
# ╔═╡ e93c5882-1ef8-43f6-b1ee-ee23c813c91b
begin
- # import Pkg
- # Pkg.activate(mktempdir())
- # Pkg.add([
- # Pkg.PackageSpec(name="ImageIO", version="0.5"),
- # Pkg.PackageSpec(name="ImageShow", version="0.3"),
- # Pkg.PackageSpec(name="FileIO", version="1.9"),
- # Pkg.PackageSpec(name="CommonMark", version="0.8"),
- # Pkg.PackageSpec(name="Plots", version="1.16"),
- # Pkg.PackageSpec(name="PlotThemes", version="2.0"),
- # Pkg.PackageSpec(name="LaTeXStrings", version="1.2"),
- # Pkg.PackageSpec(name="PlutoUI", version="0.7"),
- # Pkg.PackageSpec(name="Pluto", version="0.14"),
- # Pkg.PackageSpec(name="SymPy", version="1.0"),
- # Pkg.PackageSpec(name="HypertextLiteral", version="0.7"),
- # Pkg.PackageSpec(name="ImageTransformations", version="0.8")
- # ])
-
- using CommonMark, ImageIO, FileIO, ImageShow
- using PlutoUI
- using Plots, PlotThemes, LaTeXStrings, Random
- using SymPy
- using HypertextLiteral
- using ImageTransformations
- using Dates
- using PrettyTables
+ # import Pkg
+ # Pkg.activate(mktempdir())
+ # Pkg.add([
+ # Pkg.PackageSpec(name="ImageIO", version="0.5"),
+ # Pkg.PackageSpec(name="ImageShow", version="0.3"),
+ # Pkg.PackageSpec(name="FileIO", version="1.9"),
+ # Pkg.PackageSpec(name="CommonMark", version="0.8"),
+ # Pkg.PackageSpec(name="Plots", version="1.16"),
+ # Pkg.PackageSpec(name="PlotThemes", version="2.0"),
+ # Pkg.PackageSpec(name="LaTeXStrings", version="1.2"),
+ # Pkg.PackageSpec(name="PlutoUI", version="0.7"),
+ # Pkg.PackageSpec(name="Pluto", version="0.14"),
+ # Pkg.PackageSpec(name="SymPy", version="1.0"),
+ # Pkg.PackageSpec(name="HypertextLiteral", version="0.7"),
+ # Pkg.PackageSpec(name="ImageTransformations", version="0.8")
+ # ])
+
+ using CommonMark, ImageIO, FileIO, ImageShow
+ using PlutoUI
+ using Plots, PlotThemes, LaTeXStrings, Random
+ using SymPy
+ using HypertextLiteral
+ using ImageTransformations
+ using Dates
+ using PrettyTables
end
# ╔═╡ 69d7b791-2e69-490c-8d10-10fa433f0a72
@@ -125,44 +125,44 @@ where ``i`` is the __index of summation__, ``a_i`` is the th __``i``th term__ of
"""
# ╔═╡ 0edc99ec-c39d-4a9e-af0d-c9778c6b4211
-begin
- hline = html"
"
-md"""
-#### Summation Properties
-
-```math
-
-\begin{array}{lcl}
- \displaystyle\sum_{i=1}^n c a_i &=& c\sum_{i=1}^n a_i \\
-\\
- \displaystyle\sum_{i=1}^n (a_i+b_i) &=& \sum_{i=1}^n a_i+\sum_{i=1}^n b_i \\
-\\
-\displaystyle\sum_{i=1}^n (a_i-b_i) &=& \sum_{i=1}^n a_i-\sum_{i=1}^n b_i \\
-\\
-\end{array}
-```
-
-#### Summation Formulas
-
-```math
-\displaystyle
-\begin{array}{ll}
-(1) & \displaystyle\sum_{i=1}^n c = cn, \quad c \text{ is a constant} \\
-\\
-(2) & \displaystyle\sum_{i=1}^n i = \frac{n(n+1)}{2} \\
-\\
-(3) &\displaystyle \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} \\
-\\
-(4) & \displaystyle\sum_{i=1}^n i^3 = \left[\frac{n(n+1)}{2}\right]^2 \\
-\\
-\end{array}
-```
-
-
-
-$hline
-
-"""
+begin
+ hline = html""
+ md"""
+ #### Summation Properties
+
+ ```math
+
+ \begin{array}{lcl}
+ \displaystyle\sum_{i=1}^n c a_i &=& c\sum_{i=1}^n a_i \\
+ \\
+ \displaystyle\sum_{i=1}^n (a_i+b_i) &=& \sum_{i=1}^n a_i+\sum_{i=1}^n b_i \\
+ \\
+ \displaystyle\sum_{i=1}^n (a_i-b_i) &=& \sum_{i=1}^n a_i-\sum_{i=1}^n b_i \\
+ \\
+ \end{array}
+ ```
+
+ #### Summation Formulas
+
+ ```math
+ \displaystyle
+ \begin{array}{ll}
+ (1) & \displaystyle\sum_{i=1}^n c = cn, \quad c \text{ is a constant} \\
+ \\
+ (2) & \displaystyle\sum_{i=1}^n i = \frac{n(n+1)}{2} \\
+ \\
+ (3) &\displaystyle \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} \\
+ \\
+ (4) & \displaystyle\sum_{i=1}^n i^3 = \left[\frac{n(n+1)}{2}\right]^2 \\
+ \\
+ \end{array}
+ ```
+
+
+
+ $hline
+
+ """
end
# ╔═╡ 164b1c78-9f7b-4f9d-a6a6-fbe754cdb43e
@@ -203,70 +203,70 @@ and the $x$-axis between $x=0$ and $x=2$.
"""
# ╔═╡ 8ad65bee-9135-11eb-166a-837031c4bc45
-f(x)=5-x^2
+f(x) = 5 - x^2
# ╔═╡ e7a87684-49b0-428c-9fef-248cf868cf33
-begin
- ns = @bind n Slider(2:4000,show_value=true, default=4)
- as = @bind a NumberField(0:1)
- bs = @bind b NumberField(a+2:10)
- lrs = @bind lr Select(["l"=>"Left","r"=>"Right","m"=>"Midpoint","rnd"=>"Random"])
-
- md"""
- n = $ns a = $as b = $bs method = $lrs
-
- """
+begin
+ ns = @bind n Slider(2:4000, show_value=true, default=4)
+ as = @bind a NumberField(0:1)
+ bs = @bind b NumberField(a+2:10)
+ lrs = @bind lr Select(["l" => "Left", "r" => "Right", "m" => "Midpoint", "rnd" => "Random"])
+
+ md"""
+ n = $ns a = $as b = $bs method = $lrs
+
+ """
end
# ╔═╡ 74f6ac5d-f974-4ea6-801c-b88fe3346e55
@bind showPlot Radio(["show" => "✅", "hide" => "❌"], default="hide")
# ╔═╡ c894d994-a7fc-4e07-8941-e9f9aa89fef0
-begin
- if showPlot=="show"
- Δx =(b-a)/n
- xx1 =a:0.1:b
-
- # plot(f;xlim=(-2π,2π), xticks=(-2π:(π/2):2π,["$c π" for c in -2:0.5:2]))
-
- # recs= [rect(sample(p,Δx),Δx,p,f) for p in partition]
- # pp1=plot(xx1,f.(xx1);legend=nothing)
- pp1 = plot(xx1, f.(xx1), fillrange = zero, fillalpha = 0.35, c = :blue, framestyle=:origin, label=nothing)
- anck1 = (b-a)/2
- anck2 = f(anck1)/2
- annotate!(pp1,[(anck1,anck2,L"$S$",12)])
- annotate!(pp1,[(anck1,f(anck1),L"$y=%$f(x)$",12)])
- end
+begin
+ if showPlot == "show"
+ Δx = (b - a) / n
+ xx1 = a:0.1:b
+
+ # plot(f;xlim=(-2π,2π), xticks=(-2π:(π/2):2π,["$c π" for c in -2:0.5:2]))
+
+ # recs= [rect(sample(p,Δx),Δx,p,f) for p in partition]
+ # pp1=plot(xx1,f.(xx1);legend=nothing)
+ pp1 = plot(xx1, f.(xx1), fillrange=zero, fillalpha=0.35, c=:blue, framestyle=:origin, label=nothing)
+ anck1 = (b - a) / 2
+ anck2 = f(anck1) / 2
+ annotate!(pp1, [(anck1, anck2, L"$S$", 12)])
+ annotate!(pp1, [(anck1, f(anck1), L"$y=%$f(x)$", 12)])
+ end
end
# ╔═╡ 2da325ba-48cc-44b3-be34-e0cb46e33068
@bind showConnc Radio(["show" => "✅", "hide" => "❌"], default="hide")
# ╔═╡ 8436d1b3-c03e-42e6-bbff-e785738e0f89
-(showConnc=="show") ? md"""
-$$A=\lim_{n\to \infty} R_n =\lim_{n\to \infty} L_n =\frac{22}{3}$$
-""" : ""
+(showConnc == "show") ? md"""
+ $$A=\lim_{n\to \infty} R_n =\lim_{n\to \infty} L_n =\frac{22}{3}$$
+ """ : ""
# ╔═╡ d00038ba-98e9-45db-91df-dc75cb8ec101
begin
- findingAreaP = plot(0.2:0.1:4, x->0.6x^3-(10/3)*x^2+(13/3)*x+1.4, fillrange = zero, fillalpha = 0.35, c = :red, framestyle=:origin, label=nothing,ticks=nothing)
- plot!(findingAreaP,-0.1:0.1:4.1, x->0.6x^3-(10/3)*x^2+(13/3)*x+1.4,c=:green,label=nothing)
- annotate!(findingAreaP, [
- (0.1,4,text(L"y",14)),
- (4.1,0.1,text(L"x",14)),
- (0.2,-0.1,text(L"a",14)),
- (4,-0.1,text(L"b",14)),
- (3.9,4,text(L"f",14))
- ])
-cm"""
-### Finding Area by the Limit Definition
-
-__Find the area of the region is bounded below by the ``x``-axis, and the left and right boundaries of the region are the vertical lines ``x=a`` and ``x=b``.__
-
-$findingAreaP
-
-$(Resource("https://www.dropbox.com/s/hnspiptmyybneqn/area_with_lower_and_upper.jpg?raw=1",:width=>400))
-"""
+ findingAreaP = plot(0.2:0.1:4, x -> 0.6x^3 - (10 / 3) * x^2 + (13 / 3) * x + 1.4, fillrange=zero, fillalpha=0.35, c=:red, framestyle=:origin, label=nothing, ticks=nothing)
+ plot!(findingAreaP, -0.1:0.1:4.1, x -> 0.6x^3 - (10 / 3) * x^2 + (13 / 3) * x + 1.4, c=:green, label=nothing)
+ annotate!(findingAreaP, [
+ (0.1, 4, text(L"y", 14)),
+ (4.1, 0.1, text(L"x", 14)),
+ (0.2, -0.1, text(L"a", 14)),
+ (4, -0.1, text(L"b", 14)),
+ (3.9, 4, text(L"f", 14))
+ ])
+ cm"""
+ ### Finding Area by the Limit Definition
+
+ __Find the area of the region is bounded below by the ``x``-axis, and the left and right boundaries of the region are the vertical lines ``x=a`` and ``x=b``.__
+
+ $findingAreaP
+
+ $(Resource("https://www.dropbox.com/s/hnspiptmyybneqn/area_with_lower_and_upper.jpg?raw=1",:width=>400))
+ """
end
# ╔═╡ ef203912-b238-40a7-9d1b-4ed9b86ccbd2
@@ -317,12 +317,12 @@ $(Resource("https://www.dropbox.com/s/a3sjz8m9vspp5ec/area_def.jpg?raw=1"))
# ╔═╡ 1081bd99-7658-4c32-812c-14235bd82596
begin
- cm"""
- __Example__
-
- Find the area of the region bounded by the graph of ``f(x)=x^3`` , the ``x``-axis, and the vertical lines ``x=0`` and ``x=1``.
+ cm"""
+ __Example__
- """
+ Find the area of the region bounded by the graph of ``f(x)=x^3`` , the ``x``-axis, and the vertical lines ``x=0`` and ``x=1``.
+
+ """
end
# ╔═╡ c97d5915-7f1f-4fd6-80d3-aecb256ea0de
@@ -358,19 +358,19 @@ md""" ## Section 5.3
"""
# ╔═╡ d854d0ea-c5dd-4efa-9f46-83807339e163
-g(x)=√x
+g(x) = √x
# ╔═╡ bceda6d4-b93f-4282-8f03-fc44132ea1bb
-begin
- ns2 = @bind n2 Slider(2:2000,show_value=true, default=4)
- as2 = @bind a2 NumberField(-10:10, default=0)
- bs2 = @bind b2 NumberField(a+1:10)
- lrs2 = @bind lr2 Select(["l"=>"Left","r"=>"Right","m"=>"Midpoint", "rnd"=>"Random"])
- md"""
- n = $ns2 a = $as2 b = $bs2 method = $lrs2
-
-
- """
+begin
+ ns2 = @bind n2 Slider(2:2000, show_value=true, default=4)
+ as2 = @bind a2 NumberField(-10:10, default=0)
+ bs2 = @bind b2 NumberField(a+1:10)
+ lrs2 = @bind lr2 Select(["l" => "Left", "r" => "Right", "m" => "Midpoint", "rnd" => "Random"])
+ md"""
+ n = $ns2 a = $as2 b = $bs2 method = $lrs2
+
+
+ """
end
# ╔═╡ 7a4f6354-3d0c-4814-8c4c-2d2200568545
@@ -492,19 +492,19 @@ md"""
# ╔═╡ e427ab16-9d5a-4200-8d96-8e49ec0da312
begin
- f2(x) = sin(x)+2
- theme(:wong)
- x = 1:0.1:5
- y = f2.(x)
- p3=plot(x,y, label=nothing)
- plot!(p3,x,y/2,ribbon=y/2, linestyle=:dot,linealpha=0.1, framestyle=:origin, xticks=(1:5,[:a,"","","",:b]), label=nothing, ylims=(-1,4))
- annotate!(p3,[(3.5,2.5,L"y=f(x)"),(5.2,0,L"x"),(0.2,4,L"y")])
- # annotate!(p2,[(4,0.51,(L"$\sum_{i=1}^{%$n2} f (x^*_{i})\Delta x=%$s2$",12))])
-
- md""" * If ``f(x)\ge 0``, the integral ``\int_a^b f(x) dx`` is the area under the curve ``y=f(x)`` from ``a`` to ``b``.
-
- $p3
- """
+ f2(x) = sin(x) + 2
+ theme(:wong)
+ x = 1:0.1:5
+ y = f2.(x)
+ p3 = plot(x, y, label=nothing)
+ plot!(p3, x, y / 2, ribbon=y / 2, linestyle=:dot, linealpha=0.1, framestyle=:origin, xticks=(1:5, [:a, "", "", "", :b]), label=nothing, ylims=(-1, 4))
+ annotate!(p3, [(3.5, 2.5, L"y=f(x)"), (5.2, 0, L"x"), (0.2, 4, L"y")])
+ # annotate!(p2,[(4,0.51,(L"$\sum_{i=1}^{%$n2} f (x^*_{i})\Delta x=%$s2$",12))])
+
+ md""" * If ``f(x)\ge 0``, the integral ``\int_a^b f(x) dx`` is the area under the curve ``y=f(x)`` from ``a`` to ``b``.
+
+ $p3
+ """
end
@@ -515,45 +515,45 @@ $(load(download("https://www.dropbox.com/s/ol9l38j2a53usei/note3.png?raw=1")))
"""
# ╔═╡ 2bef2339-7afe-427d-bdc5-19b9e9b43878
-begin
- s52q1Check = @bind s52q1chk Radio(["show"=>"show","hide"=>"hide"],default="hide")
- q1Img = download("https://www.dropbox.com/s/7esby3czioyzk26/q1.png?dl=0")
-md"""
-**Question 1:**
+begin
+ s52q1Check = @bind s52q1chk Radio(["show" => "show", "hide" => "hide"], default="hide")
+ q1Img = download("https://www.dropbox.com/s/7esby3czioyzk26/q1.png?dl=0")
+ md"""
+ **Question 1:**
-$(load(q1Img))
+ $(load(q1Img))
-where each of the regions ``A, B`` and ``C`` has area equal to 5, then the area between the graph and the x-axis from ``x=-4`` to ``x=2`` is
+ where each of the regions ``A, B`` and ``C`` has area equal to 5, then the area between the graph and the x-axis from ``x=-4`` to ``x=2`` is
-$(s52q1Check)
-
-"""
+ $(s52q1Check)
+
+ """
end
# ╔═╡ 0f3814d4-6ee7-4242-88ea-5ecc7bf752bf
-md" the nswer is = **$((s52q1chk ==\"show\") ? 15 : \"\")**"
+md" the nswer is = **$((s52q1chk ==\"show\") ? 15 : \"\")**"
# ╔═╡ 05eb2a4e-2552-4bed-9523-d4f4c8760c94
-begin
- s52q1Check1 = @bind s52q1chk1 Radio(["show"=>"show","hide"=>"hide"],default="hide")
-
-md"""
-**Question 2:**
+begin
+ s52q1Check1 = @bind s52q1chk1 Radio(["show" => "show", "hide" => "hide"], default="hide")
-$(load(q1Img))
+ md"""
+ **Question 2:**
-where each of the regions ``A, B`` and ``C`` has area equal to 5, then
- ``\int_{-4}^2 f(x) dx = ``
+ $(load(q1Img))
-$(s52q1Check1)
-
-"""
+ where each of the regions ``A, B`` and ``C`` has area equal to 5, then
+ ``\int_{-4}^2 f(x) dx = ``
+
+ $(s52q1Check1)
+
+ """
end
# ╔═╡ 311050cc-9f52-43e0-afca-66d225c837d2
-md" the nswer is = **$((s52q1chk1 ==\"show\") ? -5 : \"\")**"
+md" the nswer is = **$((s52q1chk1 ==\"show\") ? -5 : \"\")**"
# ╔═╡ f5f43417-abcd-4b20-a9ff-be06157b4a02
html""
@@ -634,25 +634,25 @@ md"""
"""
# ╔═╡ 0cfb00ed-60fe-4ebb-b5e2-6182ace7a719
-begin
- xx = symbols("xx",real=true)
- sol = integrate(exp(xx),(xx,1,3))
-md"""
-#### 2. Using a Computer Algebra System
+begin
+ xx = symbols("xx", real=true)
+ sol = integrate(exp(xx), (xx, 1, 3))
+ md"""
+ #### 2. Using a Computer Algebra System
-**Example:**
+ **Example:**
-1. Set up an expression for $\int_1^3 e^x dx$ as a limit of sums.
-2. Use a computer algebra system to evaluate the expression
-
-**Solution:**
-1. In class
-"""
+ 1. Set up an expression for $\int_1^3 e^x dx$ as a limit of sums.
+ 2. Use a computer algebra system to evaluate the expression
+
+ **Solution:**
+ 1. In class
+ """
end
# ╔═╡ bfd46851-772d-43d4-8875-7d5c5dfb1155
-integrate(exp(xx),(xx,1,3))
+integrate(exp(xx), (xx, 1, 3))
# ╔═╡ 19b11522-d11c-4fe1-8f74-5dc975d82bc0
md"""
@@ -670,11 +670,11 @@ Evaluate the following integrals by interpreting each in terms of areas
# ╔═╡ 44c9faca-efb6-493c-b751-9fd69e89ecb4
begin
- f1(x)=sqrt(9-x^2)
- f3(x)=abs(x)
- theme(:wong)
- pp =plot(f1,xlims=[-4,4],ylims=[-4,4], framestyle=:origin, xtick=-4:1:4,yticks=-4:1:4)
- md"$pp"
+ f1(x) = sqrt(9 - x^2)
+ f3(x) = abs(x)
+ theme(:wong)
+ pp = plot(f1, xlims=[-4, 4], ylims=[-4, 4], framestyle=:origin, xtick=-4:1:4, yticks=-4:1:4)
+ md"$pp"
end
@@ -701,10 +701,10 @@ with $n=5$.
"""
# ╔═╡ 8d474b8c-7f6f-4ee4-9282-5e8aa0a2f7b0
-m5 = [0.2*(1/x) for x in 1.1:0.2:1.9] |> sum
+m5 = [0.2 * (1 / x) for x in 1.1:0.2:1.9] |> sum
# ╔═╡ f9e82107-07b9-4697-88fe-81b019640e6a
-integrate(1/xx,(xx,1,2)).n()
+integrate(1 / xx, (xx, 1, 2)).n()
# ╔═╡ be9f84d5-3c65-4ceb-8767-3fdc41429e12
md""" **Example**
@@ -718,7 +718,7 @@ Estimate
# ╔═╡ cf3bce53-0260-403c-8910-b04b05b558fe
begin
- exact = integrate(exp(-xx^2),(xx,0,1)).n()
+ exact = integrate(exp(-xx^2), (xx, 0, 1)).n()
end
# ╔═╡ e3d540a3-7da5-4ef6-aa31-e629e752484e
@@ -800,47 +800,47 @@ $(" ")
# ╔═╡ b592499b-cf96-486e-9067-9c79b5894641
begin
- theme(:wong)
- s54e3_f(x) = 1/x
- s54e3_x = 1:0.1:exp(1)
- s54e3_p=plot(s54e3_x,s54e3_f.(s54e3_x), label=nothing,c=:green)
- plot!(s54e3_p,s54e3_x,s54e3_f.(s54e3_x)/2,ribbon=s54e3_f.(s54e3_x)/2, linestyle=:dot,linealpha=0.1, framestyle=:origin, xticks=(1:4,[:1,:2,:3]), label=nothing, ylims=(-0.1,1.5),xlims=(-0.1,3))
- annotate!(s54e3_p,[(2,1,L"y=\frac{1}{x}"),(exp(1),-0.1,L"e")])
- cm"""
-
- __Example__
-
- Find the area of the region bounded by the graph of
- ```math
- y=\frac{1}{x}
- ```
- the ``x``-axis, and the vertical lines ``x=1`` and ``x=e``.
-
-
- $s54e3_p
-
- """
+ theme(:wong)
+ s54e3_f(x) = 1 / x
+ s54e3_x = 1:0.1:exp(1)
+ s54e3_p = plot(s54e3_x, s54e3_f.(s54e3_x), label=nothing, c=:green)
+ plot!(s54e3_p, s54e3_x, s54e3_f.(s54e3_x) / 2, ribbon=s54e3_f.(s54e3_x) / 2, linestyle=:dot, linealpha=0.1, framestyle=:origin, xticks=(1:4, [:1, :2, :3]), label=nothing, ylims=(-0.1, 1.5), xlims=(-0.1, 3))
+ annotate!(s54e3_p, [(2, 1, L"y=\frac{1}{x}"), (exp(1), -0.1, L"e")])
+ cm"""
+
+ __Example__
+
+ Find the area of the region bounded by the graph of
+ ```math
+ y=\frac{1}{x}
+ ```
+ the ``x``-axis, and the vertical lines ``x=1`` and ``x=e``.
+
+
+ $s54e3_p
+
+ """
end
# ╔═╡ bfbb3b72-dedc-476d-a028-997e98b61ae4
begin
- cm"""
- ### The Mean Value Theorem for Integrals
-
- __Theorem__ *Mean Value Theorem for Integrals*
-
- If ``f`` is continuous on the closed interval ``[a,b]``, then there exists a number ``c`` in the closed interval ``[a,b]`` such that
- ```math
- \int_a^b f(x) dx =f(c)(b-a).
- ```
-
-
+ cm"""
+ ### The Mean Value Theorem for Integrals
- $(Resource("https://www.dropbox.com/s/7fnr2kfq082kq0y/mvt.jpg?raw=1",
- :style=>"display:flex;align-items:center;flex-direction: column;"))
-
-
- """
+ __Theorem__ *Mean Value Theorem for Integrals*
+
+ If ``f`` is continuous on the closed interval ``[a,b]``, then there exists a number ``c`` in the closed interval ``[a,b]`` such that
+ ```math
+ \int_a^b f(x) dx =f(c)(b-a).
+ ```
+
+
-
- **Substitution Rule says:** It is permissible to operate with ``dx`` and ``du`` after integral signs as if they were differentials.
-
- **Example**
- Find
- ```math
- \begin{array}{ll}
- (i) & \int \bigl(x^2+1 \bigr)^2 (2x) dx \\ \\
- (ii) & \int 5e^{5x} dx \\ \\
- (iii) & \int \frac{x}{\sqrt{1-4x^2}} dx \\ \\
- (iv) & \int \sqrt{1+x^2} \;\; x^5 dx \\ \\
- (v) & \int \tan x dx \\ \\
- \end{array}
- ```
-
-
- """
+ f155(x) = x / sqrt(1 - 4 * x^2)
+ # ex1_55=plot(-0.49:0.01:0.49,f155.(-0.49:0.01:0.49), framestyle=:origin)
+ cm"""
+ __Theorem__ *Antidifferentiation of a Composite Function*
+ Let ``g`` be a function whose range is an interval ``I``, and let ``f`` be a function that is continuous on ``I``. If ``g`` is differentiable on its domain and ``F`` is an antiderivative of ``f`` on ``I``, then
+ ```math
+ \int f(g(x))g'(x)dx = F(g(x)) + C.
+ ```
+ Letting ``u=g(x)`` gives ``du=g'(x)dx`` and
+ ```math
+ \int f(u) du = F(u) + C.
+ ```
+
+