In [1]:

    

versioninfo()


    

    




Julia Version 1.4.1
Commit 381693d3df* (2020-04-14 17:20 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: Intel(R) Core(TM) i5-7200U CPU @ 2.50GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
Environment:
  JULIA_NUM_THREADS = 4

In [2]:

    

# Ovu naredbu je potrebno izvršiti sam jednom, učita se paket SymPy
using SymPy


    

In [3]:

    

# Ova naredba daje popis svih djelova paketa, uglavnom su to funkcije. Klik na lijevoj strani otvara scroll prozor, 
# drugi klik ponovo pokazuje sve. Dvostruki klik kolabira izalaz.
varinfo(SymPy)


    

    
Out[3]:






name
size
summary




@symfuns
0 bytes
SymPy.var"#@symfuns"


@syms
0 bytes
SymPy.var"#@syms"


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SymPy.var"#@vars"


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0 bytes
typeof(And)


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0 bytes
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0 bytes
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Equality
0 bytes
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False
16 bytes
Sym


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0 bytes
typeof(Ge)


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0 bytes
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0 bytes
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Heaviside
0 bytes
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IM
16 bytes
Sym


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0 bytes
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0 bytes
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Lt
40 bytes
UnionAll


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0 bytes
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0 bytes
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0 bytes
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0 bytes
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Sym
192 bytes
DataType


SymFunction
200 bytes
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192 bytes
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192 bytes
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192 bytes
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SymPy
840.453 KiB
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In [4]:

    

# Izracunajmo nekoliko integrala. Prikazuje se samo zadnji rezultat prozora
x=Sym("x")
integrate(x^2*sin(x))


    

    
Out[4]:





\begin{equation*}- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}\end{equation*}

In [5]:

    

integrate(exp(-x^2))


    

    
Out[5]:





\begin{equation*}\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\end{equation*}

In [6]:

    

# Primjer rekurzivne formule. Ovo traje malo duže
integrate(1/(1+x^2)^13)


    

    
Out[6]:





\begin{equation*}\frac{334639305 x^{23} + 3904125225 x^{21} + 20814564771 x^{19} + 67013911107 x^{17} + 144986993866 x^{15} + 221803567050 x^{13} + 245588699190 x^{11} + 197767334710 x^{9} + 114444262845 x^{7} + 46038446685 x^{5} + 12013258455 x^{3} + 1741541175 x}{2076180480 x^{24} + 24914165760 x^{22} + 137027911680 x^{20} + 456759705600 x^{18} + 1027709337600 x^{16} + 1644334940160 x^{14} + 1918390763520 x^{12} + 1644334940160 x^{10} + 1027709337600 x^{8} + 456759705600 x^{6} + 137027911680 x^{4} + 24914165760 x^{2} + 2076180480} + \frac{676039 \operatorname{atan}{\left(x \right)}}{4194304}\end{equation*}

In [7]:

    

# Kraća rekurzivna formula
integrate(1/(1+x^2)^3)


    

    
Out[7]:





\begin{equation*}\frac{3 x^{3} + 5 x}{8 x^{4} + 16 x^{2} + 8} + \frac{3 \operatorname{atan}{\left(x \right)}}{8}\end{equation*}

In [8]:

    

methods(diff)


    

    
Out[8]:




# 6 methods for generic function diff:
 diff(a::SparseArrays.AbstractSparseMatrixCSC; dims) in SparseArrays at C:\Users\Ivan_Slapnicar\AppData\Local\Programs\Julia\Julia-1.4.1\share\julia\stdlib\v1.4\SparseArrays\src\linalg.jl:1047
 
 diff(a::AbstractArray{T,1} where T) in Base at multidimensional.jl:793
 
 diff(a::AbstractArray{T,N}; dims) where {T, N} in Base at multidimensional.jl:826
 
 diff(f::Function) in SymPy at C:\Users\Ivan_Slapnicar\.julia\packages\SymPy\JaxDZ\src\mathfuns.jl:57
 
 diff(f::Function, n::Int64) in SymPy at C:\Users\Ivan_Slapnicar\.julia\packages\SymPy\JaxDZ\src\mathfuns.jl:57
 
 diff(ex::SymPy.SymbolicObject, args...; kwargs...) in SymPy at C:\Users\Ivan_Slapnicar\.julia\packages\SymPy\JaxDZ\src\importexport.jl:70
 

In [9]:

    

# Derivacija
diff(x^2*exp(sin(1/x)))


    

    
Out[9]:





\begin{equation*}2 x e^{\sin{\left(\frac{1}{x} \right)}} - e^{\sin{\left(\frac{1}{x} \right)}} \cos{\left(\frac{1}{x} \right)}\end{equation*}

In [10]:

    

# Racionanla funkcija trigonometrijskih funkcija (univerzalna trigonometrijska supstitucija)
integrate(1/((2+cos(x))*sin(x)))


    

    
Out[10]:





\begin{equation*}\frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 3 \right)}}{3} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{3}\end{equation*}

In [12]:

    

integrate(1/(4*sin(x)+3*cos(x)+5))


    

    
Out[12]:





\begin{equation*}- \frac{1}{\tan{\left(\frac{x}{2} \right)} + 2}\end{equation*}

In [13]:

    

integrate(1/(sin(x)*(2+cos(x)-2*sin(x))))


    

    
Out[13]:





\begin{equation*}\frac{5 \log{\left(\tan{\left(\frac{x}{2} \right)} - 3 \right)}}{3} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{3}\end{equation*}

In [ ]:

    

# Ovo traje beskonačno?
# integrate(cos(x)^3/(sin(x)^2+sin(x)))


    

In [14]:

    

# Ovo je isti zadatak kao gore, ali traje kratko. Koristi se jednostavnija supstitucija. 
integrate((1-sin(x)^2)*cos(x)/(sin(x)^2+sin(x)))


    

    
Out[14]:





\begin{equation*}\log{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)}\end{equation*}

In [15]:

    

# Ovo traje jako dugo i ne uspije izračunati nego vrati polazni integral
# integrate( (2*tan(x)+3)/(sin(x)^2+2*cos(x)^2))


    

In [16]:

    

t=Sym("t")
I₁=integrate( (2*t+3)/(t^2+2))


    

    
Out[16]:





\begin{equation*}\log{\left(t^{2} + 2 \right)} + \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} t}{2} \right)}}{2}\end{equation*}

In [18]:

    

# Vratimo supstituciju natrag
I₂=convert(Function,I₁)
I₂(tan(x))


    

    
Out[18]:





\begin{equation*}\log{\left(\tan^{2}{\left(x \right)} + 2 \right)} + 2.12132034355964 \operatorname{atan}{\left(0.707106781186548 \tan{\left(x \right)} \right)}\end{equation*}

In [19]:

    

# Ovo traje beskonačno!
# I₄=integrate((1+sinh(x))/((2+cosh(x))*(3+sinh(x))),x)
# Program treba našu pomoć!!!


    

In [20]:

    

I₄=(1+sinh(x))/((2+cosh(x))*(3+sinh(x)))


    

    
Out[20]:





\begin{equation*}\frac{\sinh{\left(x \right)} + 1}{\left(\sinh{\left(x \right)} + 3\right) \left(\cosh{\left(x \right)} + 2\right)}\end{equation*}

In [22]:

    

# supstitucija za sinh(x)
I₅=subs(I₄,sinh(x),(2*t)/(1-t^2))


    

    
Out[22]:





\begin{equation*}\frac{\frac{2 t}{1 - t^{2}} + 1}{\left(\frac{2 t}{1 - t^{2}} + 3\right) \left(\cosh{\left(x \right)} + 2\right)}\end{equation*}

In [23]:

    

# Supstitucija za cosh(x)
I₆=subs(I₅,cosh(x),(1+t^2)/(1-t^2))


    

    
Out[23]:





\begin{equation*}\frac{\frac{2 t}{1 - t^{2}} + 1}{\left(2 + \frac{t^{2} + 1}{1 - t^{2}}\right) \left(\frac{2 t}{1 - t^{2}} + 3\right)}\end{equation*}

In [24]:

    

# Pomnožimo s dx
I₇=I₆*2/(1-t^2)


    

    
Out[24]:





\begin{equation*}\frac{2 \left(\frac{2 t}{1 - t^{2}} + 1\right)}{\left(1 - t^{2}\right) \left(2 + \frac{t^{2} + 1}{1 - t^{2}}\right) \left(\frac{2 t}{1 - t^{2}} + 3\right)}\end{equation*}

In [25]:

    

# Integriramo racionalnu funkciju 
I₈=integrate(I₇,t)


    

    
Out[25]:





\begin{equation*}\frac{\log{\left(t^{2} - 3 \right)}}{3} - \left(\frac{1}{3} - \frac{\sqrt{10}}{15}\right) \log{\left(t - \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} - \left(\frac{\sqrt{10}}{15} + \frac{1}{3}\right) \log{\left(t - \frac{\sqrt{10}}{3} - \frac{1}{3} \right)}\end{equation*}

In [26]:

    

# Vratimo supstituciju natrag
I₉=subs(I₈,t,tanh(x/2))


    

    
Out[26]:





\begin{equation*}\frac{\log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} - 3 \right)}}{3} - \left(\frac{1}{3} - \frac{\sqrt{10}}{15}\right) \log{\left(\tanh{\left(\frac{x}{2} \right)} - \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} - \left(\frac{\sqrt{10}}{15} + \frac{1}{3}\right) \log{\left(\tanh{\left(\frac{x}{2} \right)} - \frac{\sqrt{10}}{3} - \frac{1}{3} \right)}\end{equation*}

In [ ]: