Задачи върху неопределени интеграли

Интеграли - 1 част
Интеграли - 2 част
Интеграли - 3 част
Интеграли - 4 част

Интеграли Съдържащи sec ax

$\int_{}^{}secaxdx=\frac{1}{a}.ln(secax+tanax)=\left(\frac{1}{a}\right)ln(tan\left(\frac{\pi}{4}+\frac{ax}{2}\right))$

$\int_{}^{}sec^2axdx=\frac{tanax}{a}$

$\int_{}^{}sec^3axdx=\frac{secax.tanax}{2a}+\frac{1}{2a}.ln(secax+tanax)$

$\int_{}^{}sec^nax.tanaxdx=\frac{sec^nax}{na}$

$\int_{}^{}\frac{dx}{secax}=\frac{sinax}{a}$

$\int_{}^{}x.sec^2axdx=\frac{x}{a}.tanax+\frac{1}{a^2}.ln(cosax)$

Интеграли Съдържащи csc ax

$\int_{}^{}cscaxdx=\frac{1}{a}.ln(cscax-cotax)=\frac{1}{a}ln(tan\left(\frac{ax}{2}\right))$

$\int_{}^{}csc^2axdx=\frac{-cotax}{a}$

$\int_{}^{}csc^3axdx=\frac{-cscax.cotax}{2a}+\frac{1}{2a}.ln(tan\left(\frac{ax}{2}\right))$

$\int_{}^{}csc^nax.cotaxdx=\frac{-csc^nax}{na}$

$\int_{}^{}\frac{dx}{cscax}=\frac{-cosax}{a}$

$\int_{}^{}x.csc^2axdx=\frac{-x}{a}.cotax+\frac{1}{a^2}.ln(sinax)$

Интеграли Съдържащи Обратни Тригонометрични Функции

$\int_{}^{}sin^{-1}\frac{x}{a}dx=x.sin^{-1}\frac{x}{a}+\sqrt{a^2-x^2}$

$\int_{}^{}x.sin^{-1}\frac{x}{a}dx=\left(\frac{x^2}{2}-\frac{a^2}{4}\right)sin^{-1}\frac{x}{a}+\frac{x}{4}\sqrt{a^2-x^2}$

$\int_{}^{}x^2.sin^{-1}\frac{x}{a}dx=\frac{x^3}{3}sin^{-1}\frac{x}{a}+\frac{x^2+2a^2}{9}\sqrt{a^2-x^2}$

$\int_{}^{}\frac{sin^{-1}\frac{x}{a}}{x}dx=\frac{\left(\frac{x}{a}\right)^3}{2*3*3}+\frac{1*3\left(\frac{x}{a}\right)^5}{2*4*5*5}+\frac{1*3*5\left(\frac{x}{a}\right)^7}{2*4*6*7*7}+...$

$\int_{}^{}\frac{sin^{-1}\frac{x}{a}}{x^2}dx=\frac{-sin^{-1}\frac{x}{a}}{x}-\frac{1}{a}ln\left(a+\frac{\sqrt{a^2-x^2}}{x}\right)$

$\int_{}^{}\left(sin^{-1}\frac{x}{a}\right)^2dx=x\left(sin^{-1}\frac{x}{a}\right)^2-2x+2\sqrt{a^2-x^2}.sin^{-1}\frac{x}{a}$

$\int_{}^{}cos^{-1}\frac{x}{a}dx=x.cos^{-1}\frac{x}{a}-\sqrt{a^2-x^2}$

$\int_{}^{}x.cos^{-1}\frac{x}{a}dx=\left(\frac{x^2}{2}-\frac{a^2}{4}\right)cos^{-1}\frac{x}{a}-\frac{x}{4}\sqrt{a^2-x^2}$

$\int_{}^{}x^2.cos^{-1}\frac{x}{a}dx=\frac{x^3}{3}cos^{-1}\frac{x}{a}-\frac{x^2+2a^2}{9}\sqrt{a^2-x^2}$

$\int_{}^{}\frac{cos^{-1}\frac{x}{a}}{x}dx=\frac{\pi}{2}lnx-\int_{}^{}\frac{sin^{-1}\frac{x}{a}}{x}dx$

$\int_{}^{}\frac{cos^{-1}\frac{x}{a}}{x^2}dx=\frac{-cos^{-1}\frac{x}{a}}{x}+\frac{1}{a}ln\left(a+\frac{\sqrt{a^2-x^2}}{x}\right)$

$\int_{}^{}\left(cos^{-1}\frac{x}{a}\right)^2dx=x\left(sin^{-1}\frac{x}{a}\right)^2-2x-2\sqrt{a^2-x^2}.cos^{-1}\frac{x}{a}$

$\int_{}^{}tan^{-1}\frac{x}{a}dx=x.tan^{-1}\frac{x}{a}-\frac{a}{2}ln(x^2+a)$

$\int_{}^{}x.tan^{-1}\frac{x}{a}dx=\frac{1}{2}(x^2+a^2)tan^{-1}\frac{x}{a}-\frac{ax}{2}$

$\int_{}^{}x^2.tan^{-1}\frac{x}{a}dx=\frac{x^3}{3}tan^{-1}\frac{x}{a}-\frac{ax^2}{6}+\frac{a^3}{6}ln(x^2+a)$

$\int_{}^{}\frac{tan^{-1}\frac{x}{a}}{x}dx=\frac{x}{a}-\frac{\left(\frac{x}{a}\right)^3}{3^2}+\frac{\left(\frac{x}{a}\right)^5}{5^2}-\frac{\left(\frac{x}{a}\right)^7}{7^2}+...$

$\int_{}^{}\frac{tan^{-1}\frac{x}{a}}{x^2}dx=\frac{-1}{x}tan^{-1}\frac{x}{a}-\frac{1}{2a}ln\frac{x^2+a^2}{x^2}$

$\int_{}^{}cot^{-1}\frac{x}{a}dx=x.cot^{-1}\frac{x}{a}+\frac{a}{2}ln(x^2+a^2)$

$\int_{}^{}x.cot^{-1}\frac{x}{a}dx=\frac{1}{2}(x^2+a^2)cot^{-1}\frac{x}{a}+\frac{ax}{2}$

$\int_{}^{}x^2.cot^{-1}\frac{x}{a}dx=\frac{x^3}{3}cot^{-1}\frac{x}{a}+\frac{ax^2}{6}-\frac{a^3}{6}ln(x^2+a)$

$\int_{}^{}\frac{cot^{-1}\frac{x}{a}}{x}dx=\frac{\pi}{2}lnx-\int_{}^{}\frac{tan\frac{x}{a}}{x}dx$

$\int_{}^{}\frac{cot^{-1}\frac{x}{a}}{x^2}dx=\frac{-1}{x}cot^{-1}\frac{x}{a}+\frac{1}{2a}ln\frac{x^2+a^2}{x^2}$

Интеграли Съдържащи e^ax

$\int_{}^{}e^{ax}dx=\frac{e^{ax}}{a}$

$\int_{}^{}x.e^{ax}dx=e^{ax}\left(x-\frac{1}{a}\right)$

$\int_{}^{}x^2.e^{ax}dx=e^{ax}\left(x^2-\frac{2x}{a}+\frac{2}{a^2}\right)$

$\int_{}^{}x^n.e^{ax}dx=\frac{x^n.e^{ax}}{a}-\frac{n}{a}\int_{}^{}x^{n-1}.e^{ax}dx=\frac{e^{ax}}{a}.\left(x^n-\frac{n.x^{n-1}}{a}+\frac{n(n-1).x^{n-2}}{a^2}-...\frac{(-1)^n.n!}{a^n}\right)$ , ако n е положително число