Bosqichli kalkulyatorlar:

Bu kalkulyator \(F\left(x,\,y,\,y',\,y'',\dots,y^{\left(n\right)}\right)=0\) — turli tartibli oddiy differensial tenglamalarni (ODT) yechadi, jumladan:

Ajraladigan tenglamalar: \(p\left(x\right)\mathrm{d}x=q\left(y\right)\mathrm{d}y\)

Bir jinsli tenglamalar: \(y'=f\left(k\,x,\;k\,y\right)=f\left(x,\;y\right)\)

Birinchi tartibli chiziqli tenglamalar: \(y'+a\left(x\right)\,y=b\left(x\right)\)

Quyidagi ko'rinishdagi tenglamalar: \(y'=f\left(\frac{a_1\,x+b_1\,y+c_1}{a\,x+b\,y+c}\right)\)

Bernulli differensial tenglamalari: \(y'+a\left(x\right)\,y=b\left(x\right)\,y^n\)

Rikkati tenglamalari: \(y'+a\left(x\right)\,y+b\left(x\right)\,y^2=c\left(x\right)\)

To'liq differensial tenglamalar: \(P\left(x,\;y\right)\,\mathrm{d}x+Q\left(x,\;y\right)\,\mathrm{d}y=0\)

To'liq bo'lmagan differensial tenglamalar: \(\mu\cdot P\left(x,\;y\right)\,\mathrm{d}x+\mu\cdot Q\left(x,\;y\right)\,\mathrm{d}y=0\) — bu yerda \(\mu\) integrallovchi ko'paytiruvchi

To'liq differensial tenglamalar: \(\mathrm{d}\left(F\left(x,\,y\right)\right)=0\)

Hosilaga nisbatan yechilmagan tenglamalar: \(F\left(x,\;y,\;y'\right)=0\)

Quyidagi ko'rinishdagi tenglamalar: \(F\left(x,\,y^{\left(k\right)},\,y^{\left(k+1\right)},\dots,y^{\left(n\right)}\right)=0\) va \(F\left(y,\,y',\,y''\,\dots,y^{\left(n\right)}\right)=0\)

O'zgarmas koeffitsientli chiziqli differensial tenglamalar: \(y^{\left(n\right)}+a_{n-1}\,y^{\left(n-1\right)}+\ldots+a_0\,y=f\left(x\right)\)

Koshi-Eyler tenglamalari: \(x^n\,y^{\left(n\right)}+a_{n-1}\,x^{n-1}\,y^{\left(n-1\right)}+\ldots+a_{1}\,x\,y'+a_0\,y=0\)

Kalkulyator shuningdek oddiy differensial tenglamalar sistemalarini ham yechadi:

O'zgarmas koeffitsientli chiziqli bir jinsli sistemalar: \(X'\left(t\right)=A\,X\left(t\right)\)

O'zgarmas koeffitsientli chiziqli bir jinsli bo'lmagan sistemalar: \(X'\left(t\right)=A\,X\left(t\right)+f\left(t\right)\)

Shuningdek, boshlang'ich shartli tenglamalar va sistemalarni (boshlang'ich qiymat masalalari) ham yechadi

Bu kalkulyator \(\displaystyle \int{f\left(x\right)\;\mathrm{d}x=F\left(x\right)+C}\) — aniqmas integrallarni quyidagi usullar va texnikalar yordamida bosqichma-bosqich yechadi:

Asosiy integrallash formulalari: \(\displaystyle\int{x^n}\;\mathrm{d}x=\dfrac{x^{n+1}}{n+1}+C,\;\left(n\neq-1\right)\), \(\displaystyle\int{a^x}\;\mathrm{d}x=\dfrac{a^x}{\ln\left(a\right)}+C\)\(\dots\)

Yig'indi va ayirma qoidasi: \(\displaystyle\int{\left(u\pm v\pm w\right)}\;\mathrm{d}x=\int{u}\;\mathrm{d}x\pm\int{v}\;\mathrm{d}x\pm\int{w}\;\mathrm{d}x\)

Doimiy ko'paytuvchi qoidasi: \(\displaystyle\int{c\,f\left(x\right)}\;\mathrm{d}x=c\int{f\left(x\right)}\;\mathrm{d}x\)

O'rniga qo'yish qoidasi (u-almashtiruv): \(\displaystyle\int{f\left(x\right)}\;\mathrm{d}x=\left[\begin{array}{c}x=\varphi\left(t\right)\\\mathrm{d}x=\varphi'\left(t\right)\,\mathrm{d}t\end{array}\right]=\int{f\left(\varphi\left(t\right)\right)\,\varphi'\left(t\right)}\;\mathrm{d}t\)

Ratsional funksiyalarni integrallash: trigonometrik \(\mathrm{R}\left(\sin\left(x\right),\;\cos\left(x\right)\right)\); giperbolik \(\mathrm{R}\left(\sinh\left(x\right),\;\cosh\left(x\right)\right)\); qisman kasrlar \(\dfrac{P_k\left(x\right)}{Q_n\left(x\right)}\)

Noaniq koeffitsientlar usuli: ko'phadlarni ko'paytuvchilarga ajratish, chiziqli-kasr irratsionalliklar \(\mathrm{R}\left(x,\,\left(\dfrac{a\,x+b}{c\,x+d}\right)^{r_1,\dots,\,r_n}\right)\), Ostrogradskiy–Ermit usuli \(\displaystyle\int{\dfrac{P\left(x\right)}{Q\left(x\right)}}=\dfrac{P_2\left(x\right)}{Q_2\left(x\right)}+\int{\dfrac{P_1\left(x\right)}{Q_1\left(x\right)}}\), kvadratik ildizlarni o'z ichiga olgan integrallar \(\mathrm{R}\left(x, \sqrt{a\,x^2+b\,x+c}\right)\), to'g'ridan-to'g'ri usullar \(\displaystyle\int{\dfrac{P_n\left(x\right)}{\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}\), \(\displaystyle\int{\dfrac{P_m\left(x\right)}{\left(x-\alpha\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}\), \(\displaystyle\int{\dfrac{M\,x+N}{\left(x^2+p\,x+q\right)^n\,\sqrt{a\,x^2+b\,x+c}}}{\;\mathrm{d}x}\)

Qismlab integrallash \(\displaystyle\int{u}{\;\mathrm{d}v}=u\,v-\int{v}{\;\mathrm{d}u}\), trigonometrik va giperbolik almashtirishlar, Eyler almashtirishlari, binom differensiallarining integrallari \(\displaystyle\int{x^m\,\left(a\,x^n+b\right)^p}{\;\mathrm{d}x}\)

\(\sin^n\left(x\right)\,\cos^m\left(x\right)\) darajalarining ko'paytmalari va giperbolik funksiyalar \(\sinh^n\left(x\right)\,\cosh^m\left(x\right)\)

Standart integrallash formulalari, absolyut qiymatlarni o'z ichiga olgan integrallash, maxsus funksiyalar \(\Gamma\left(s,\,x\right)\), \(\operatorname{Ei}\left(x\right)\), \(\operatorname{li}\left(x\right)\), \(\operatorname{Si}\left(x\right)\), \(\operatorname{Ci}\left(x\right)\), \(\operatorname{Shi}\left(x\right)\), \(\operatorname{Chi}\left(x\right)\), \(\operatorname{Li_2}\left(x\right)\), \(\operatorname{S}\left(x\right)\), \(\operatorname{C}\left(x\right)\), \(\operatorname{erf}\left(x\right)\), \(\operatorname{erfi}\left(x\right)\), teskari zanjir qoidasi \(\displaystyle\int{\mathrm{d}\left(\mathrm{F}\left(x\right)\right)}\), Veyershtrass almashtiruvi (yarim burchak tangenti), Eyler formulasi \(e^{i\,x}=\cos(x)+i\,\sin(x)\)

Eksponensial, logarifmik, trigonometrik va giperbolik almashtirishlar

Algebraik almashtirishlar va soddalashtirish bilan qayta guruhlash

Bu kalkulyator \(\displaystyle\int\limits_{a}^{b}{f\left(x\right)}{\;\mathrm{d}x}\) — aniq integrallarni bosqichma-bosqich yechadi: antidifferensiallarni hisoblaydi va Analizning asosiy teoremasini qo'llaydi, simmetrik oraliqlarda juft yoki toq funksiyalar uchun simmetriya xossalaridan va davriylik xossalaridan foydalanadi

Xos bo'lmagan integrallar uchun kalkulyator cheksizlikdagi limitlarni va integrallash oralig'idagi uzilish nuqtalarida bir tomonlama limitlarni hisoblab chiqadi

Qo'llab-quvvatlanadigan matematik funksiyalar:

\(\ln\) \(\sin\) \(\cos\) \(\tan\) \(\cot\) \(\arctan\) \(\arcsin\) \(\arccos\) \(\operatorname{arccot}\) \(\sinh\) \(\cosh\) \(\tanh\) \(\coth\) \(\operatorname{sech}\) \(\operatorname{csch}\) \(\operatorname{arsinh}\) \(\operatorname{arcosh}\) \(\operatorname{artanh}\) \(\operatorname{arcoth}\) \(\operatorname{arcsec}\) \(\operatorname{arccsc}\) \(\operatorname{arsech}\) \(\operatorname{arcsch}\) \(\sec\) \(\csc\) \(\left|f\right|\)

Kalkulyator \(f\left(x\right)=0\) ko'rinishidagi tenglamalarni yechadi, jumladan:

Funksiyaning aniqlanish sohasini aniqlash \(\mathrm{dom}\left(f\right)\)

Chiziqli tenglamalar \(a\,x+b=0\)

Haqiqiy va kompleks koeffitsientli kvadrat tenglamalar \(a\,x^2+b\,x+c=0\)

\(a\,x^3+b\,x^2+b\,x+a=0\) ko'rinishidagi kub tenglamalar

Kub tenglamalar \(a\,x^3+b\,x^2+c\,x+d=0\)

\(a\,x^4+b\,x^3+c\,x^2\pm b\,x+a=0\) va \(a\,x^4+b\,x^3+c\,x^2+d\,x+\dfrac{a\,d^2}{b^2}=0\) ko'rinishidagi to'rtinchi darajali tenglamalar

Arifmetik progressiyadagi to'rtta hadning ko'paytmasi \(\left(a\,x+b\right)\,\left(a\,x+b+c\right)\,\left(a\,x+b+2\,c\right)\,\left(a\,x+b+3\,c\right)=d\)

Turli eksponensial, logarifmik, trigonometrik, giperbolik va teskari tenglamalar

To'rtinchi darajali tenglamalarni yechish uchun Ferrari usulini qo'llash \(a\,x^4+b\,x^3+c\,x^2+d\,x+e=0\)

Ratsional ildizlarni topish \(x=\dfrac{m}{n}\) va ko'paytuvchilarga ajratish \(f_1\left(x\right)\cdots f_n\left(x\right)=0\)

Asosiy trigonometrik, giperbolik va teskari tenglamalarning ma'lum yechimlari

Kompleks sonlarning ildizlarini topish \(\sqrt[n]{a+i\,b}\)

Yarim burchak tangensiga almashtirish \(\sin(x)=\dfrac{2\,t}{1+t^2}\) va \(\cos(x)=\dfrac{1-t^2}{1+t^2}\) bu yerda \(t=\tan\left(\dfrac{x}{2}\right)\)

Binom teoremasi \((a+b)^n=a^n+C^1_n\,a^{n-1}\,b+\ldots+C^{n-1}_n\,a\,b^{n-1}+b^n\)

Yig'indi va ayirmalar uchun ko'phad ayniyatlari \(x^n+y^n\), \(x^n-y^n\)

O'xshash hadlarni birlashtirish va umumiy ko'paytuvchilarni ajratib olish \(x^2+x\;\Rightarrow\; x\,(x+1)\)

Kasrlarni ko'paytma ko'rinishida yozish \(\dfrac{a}{b}=\dfrac{c}{d}\;\Rightarrow\;a\,d=b\,c\) va to'liq kvadratga keltirish \((a+b)^2+c\)

Tabiiy logarifmlarni yo'qotish uchun ikkala tomonni darajalash

Kompleks logarifmlar \(\ln\left(a+i\,b\right)\) va Eyler formulasi \(e^{i\,x}=\cos\left(x\right)+i\,\sin\left(x\right)\)

Asosiy funksional tenglamalar \(f\left(g\left(x\right)\right) = f\left(r\left(x\right)\right)\;\Rightarrow\;g\left(x\right)=r\left(x\right)\)

Bu kalkulyator \(f\left(x\right)\) yoki \(f\left(x,\,y,\,y',\dots,\,z,\,z',\dots\right)\) funksiyaning hosilasini hisoblaydi va hosilani hisoblash uchun ishlatilgan qoidalarni ko'rsatadi.

Quyidagi qoidalar aniqlangan:

\(x\), \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), \(\cot(x)\), \(e^x\), \(a^x\), \(\ln(x)\)\(\,\ldots\) ning asosiy hosilalari

O'zgarmas qoida: \((c)'=0\)

O'zgarmasga ko'paytirish qoidasi: \(\left(c\,f(x)\right)'=c\,f'(x)\)

Yig'indi qoidasi: \(\left(f(x)+g(x)\right)'=f'(x)+g'(x)\)

Ayirma qoidasi: \(\left(f(x)-g(x)\right)'=f'(x)-g'(x)\)

Daraja qoidasi: \(\left(x^n\right)'=n\,x^{n-1}\)

Ko'paytma qoidasi: \(\left(f(x)\,g(x)\right)'=f(x)\,g'(x)+g(x)\,f'(x)\)

Bo'linma qoidasi: \(\left(\dfrac{f(x)}{g(x)}\right)'=\dfrac{g(x)\,f'(x)-f(x)\,g'(x)}{\left(g(x)\right)^2}\)

Teskari qoida: \(\left(\dfrac{1}{f(x)}\right)'=\dfrac{-f'(x)}{\left(f(x)\right)^2}\)

Zanjir qoidasi: \(\left(f\left(g(x)\right)\right)'=f'_g\left(g\right)\,g'(x)\)

Absolyut qiymat: \(\left(\left|x\right|\right)'=\dfrac{x}{\left|x\right|}\)

Ishorali funksiya: \(\left(\operatorname{sgn}\left(f\right)\right)'=2\,\delta\left(x\right)\), bu yerda \(\delta\) — Dirak delta funksiyasi

Ushbu kalkulyator quyidagi xossalardan foydalanib funksiyaning limitini \(\displaystyle\lim_{x\to{a}}{f\left(x\right)}\) topadi:

Konstanta limiti \(\displaystyle\lim_{x\to{a}}C=C\)

Konstantaga ko'paytirish qoidasi \(\displaystyle\lim_{x\to{a}}k\,f(x)=k\,\lim_{x\to{a}}f(x)\)

Yig'indi va ayirma qoidasi \(\displaystyle\lim_{x\to{a}}{f\left(x\right)\pm g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}\pm\lim_{x\to{a}}{g\left(x\right)}\)

Ko'paytma qoidasi \(\displaystyle\lim_{x\to{a}}{f\left(x\right)\,g\left(x\right)}=\lim_{x\to{a}}{f\left(x\right)}\,\lim_{x\to{a}}{g\left(x\right)}\)

Bo'linma qoidasi \(\displaystyle\lim_{x\to{a}}\dfrac{f(x)}{g(x)}=\dfrac{\displaystyle\lim_{x\to{a}}f(x)}{\displaystyle\lim_{x\to{a}}g(x)}\), agar \(\displaystyle\lim_{x\to{a}}g(x)\neq 0\)

Eksponensial funksiya limiti \(\displaystyle\lim_{x\to{a}}{a^{f\left(x\right)}}=a^{\displaystyle\lim_{x\to{a}}{f\left(x\right)}}\)

Asosiy limitlar \(\displaystyle\lim_{x\to{0}}{\dfrac{\sin\left(x\right)}{x}}=1\) va \(\displaystyle\lim_{x\to{\infty}}{(1+\dfrac{1}{x})^x}=e\)

Qisish teoremasi: agar \(g\left(x\right)\leq f\left(x\right)\leq h\left(x\right)\) va \(\displaystyle\lim_{x\to{a}}g(x)=\lim_{x\to{a}}h(x)=L\;\Rightarrow\;\lim_{x\to{a}}f(x)=L\)

Lopital qoidasi: agar \(\displaystyle\lim_{x\to{a}}f(x)=0\) va \(\displaystyle\lim_{x\to{a}}g(x)=0\) (yoki ikkala limit \(\infty\) ga teng), u holda \(\displaystyle\lim_{x\to{a}}{\dfrac{f\left(x\right)}{g\left(x\right)}}=\lim_{x\to{a}}{\dfrac{f'\left(x\right)}{g'\left(x\right)}}\)

Teylor qatori \(f(x)=\sum\limits_{n=0}^{\infty}\dfrac{f^{\left(n\right)}\left(a\right)}{n!}\,\left(x-a\right)^n\)

Qo'shma ifodaga ko'paytirish, o'zgaruvchilarni almashtirish va Eyler formulasini qo'llaydi

Ikki tomonlama limitlarni \(x\to{a}\) va bir tomonlama limitlarni \(x\to{a^+}\) hisoblaydi

Ushbu kalkulyator murakkab ifoda \(f(z)\) ni algebraik shakl \(z=a+i\,b\), trigonometrik shakl \(z=r\cdot(\cos(\varphi)+i\,\sin(\varphi))\) va eksponensial shakl \(z=r\,e^{i\,\varphi}\) ga quyidagilar yordamida o'zgartiradi:

Kompleks sonning moduli: \(r=\left|a+i\,b\right|=\sqrt{a^2+b^2}\)

Kompleks sonning ildizi: \(\sqrt[n]{z}=\sqrt[n]{r}\,\left(\cos\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)+i\,\sin\left(\dfrac{\varphi+2\,\pi\,\mathrm{k}}{n}\right)\right)\)

Kompleks sonning darajasi: \(z^n=r^n\,\left(\cos\left(n\,\varphi\right)+i\,\sin\left(n\,\varphi\right)\right)\)

Kasrni qo'shma son bilan ratsionalizatsiya qilish: \(\dfrac{z}{a+i\,b}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{\left(a+i\,b\right)\cdot\left(a-i\,b\right)}\;\Rightarrow\;\dfrac{z\cdot\left(a-i\,b\right)}{a^2+b^2}\)

Kompleks logarifm: \(\operatorname{Log}\left(z\right)=\ln\left(r\right)+i\,(\varphi+2\,\pi\,\mathrm{k})\)

Kompleks logarifmning asosiy qiymati: \(\mathrm{Im}\operatorname{Log}\in(-\pi,\,\pi]\)

Trigonometrik va giperbolik ayniyatlar, masalan \(\sin\left(\alpha\pm\beta\right)=\sin\left(\alpha\right)\,\cos\left(\beta\right)\pm\cos\left(\alpha\right)\,\sin\left(\beta\right)\) yoki \(\sinh\left(i\,b\right)=i\,\sin\left(b\right)\), va Eyler formulasi \(e^{i\,\varphi}=\cos\left(\varphi\right)+i\,\sin\left(\varphi\right)\)

Ushbu kalkulyator \(\mathrm{A}\), \(\mathrm{B}\) va \(\mathrm{C}\) matritsalari bilan berilgan matritsa ifodalarini hisoblaydi

Uning funksionalligi quyidagi matritsa amallarini o'z ichiga oladi: qo'shish \(\mathrm{A}+\mathrm{B}\), ayirish \(\mathrm{A}-\mathrm{B}\), ko'paytirish \(\mathrm{C}\cdot\mathrm{B}\), determinant \(\left|\mathrm{A}\right|\), transpozitsiya \(\mathrm{B}^{\mathrm{T}}\), rang \(\operatorname{rank}\mathrm{C}\), teskari matritsa \(\mathrm{A}^{-1}\), skalarga ko'paytirish \(a\cdot\mathrm{B}\) yoki skalar qo'shish \(c+\mathrm{A}\)

Matritsa elementlarining hosilasini hisoblaydi \(\left(\mathrm{C}\right)'_x={\scriptsize\left(\begin{gathered}\left(\mathrm{a_{11}}\right)'_x&\left(\mathrm{a_{12}}\right)'_x\\\left(\mathrm{a_{21}}\right)'_x&\left(\mathrm{a_{22}}\right)'_x\end{gathered}\right)}\) yoki matritsa elementlarining integralini hisoblaydi \(\int{\mathrm{A}}{\;\mathrm{d}x}={\scriptsize\left(\begin{gathered}\int{\mathrm{a_{11}}}{\;\mathrm{d}x}&\int{\mathrm{a_{12}}}{\;\mathrm{d}x}\\\int{\mathrm{a_{21}}}{\;\mathrm{d}x}&\int{\mathrm{a_{22}}}{\;\mathrm{d}x}\end{gathered}\right)}\)

Matematik funksiyalarni \(\sin\), \(\cos\)\(\,\ldots\) matritsa elementlariga alohida-alohida qo'llaydi, masalan \(\ln\left(\mathrm{A}\right)={\scriptsize\left(\begin{gathered}\ln\left(\mathrm{a_{11}}\right)&\ln\left(\mathrm{a_{12}}\right)\\\ln\left(\mathrm{a_{21}}\right)&\ln\left(\mathrm{a_{22}}\right)\end{gathered}\right)}\)

Raqamli qiymatlarni hamda arifmetik amallar va funksiyalar kombinatsiyalarini hisoblaydi