Integrals - 1 part
Integrals - 2 part
Integrals - 3 part
Integrals - 4 part

Integrals Involving cot ax

$\int_{}^{}cotaxdx=\frac{1}{a}.ln(sinax)$

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

$\int_{}^{}cot^3axdx=\frac{-cot^2ax}{2a}-\frac{1}{a}.ln(sinax)$

$\int_{}^{}cot^nax.csc^2axdx=\frac{-cot^{n+1}ax}{(n+1)a}$

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

$\int_{}^{}\frac{dx}{cotax}=\frac{-1}{a}.ln(cosax)$

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

Integrals Involving sec ax

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

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

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

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

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

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

Integrals Involving csc ax

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

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

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

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

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

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

Integrals Involving Inverse Trigonometric Functions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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





Integrals Involving eax

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

$\int_{}^{}x.e^{ax}dx=\frac{e^{ax}}{a^2}(ax-1) = e^{ax}\left(\frac{x}{a} - \frac{1}{a^2} \right)$

$\int_{}^{}x^2.e^{ax}dx=e^{ax}\left(\frac{x^2}{a}-\frac{2x}{a^2}-\frac{2}{a^3}\right)$

$\int_{}^{}x^n.e^{ax}dx=\frac{x^n.e^{ax}}{a}-\frac{n}{a}\int_{}^{}x^{n-1}.e^{ax}dx=\frac{e^{ax}}{a}.\left(x^n-\frac{n.x^{n-1}}{a}+\frac{n(n-1).x^{n-2}}{a^2}-...\frac{(-1)^n.n!}{a^n}\right)$, if n = positive integer


Integrals Invoving ln x

$int ln x dx = x.ln x - x$

$\int_{}^{}x.lnxdx=\frac{x^2}{2}\left(lnx-\frac{1}{2}\right)$

$\int_{}^{}x^m.lnxdx=\frac{x^{m+1}}{m+1}\left(lnx-\frac{1}{m+1}\right)$

$\int_{}^{}\frac{lnx}{x}dx=\frac{1}{2}ln^2x$

$\int_{}^{}\frac{lnx}{x^2}dx=\frac{-lnx}{x}-\frac{1}{x}$

∫ ln2x dx = x.ln2x - 2x.ln x + 2x

$\int_{}^{}\frac{ln^nx}{x}dx=\frac{ln^{n+1}x}{n+1}$

$\int_{}^{}\frac{dx}{x.lnx}=ln(lnx)$

∫ lnnx dx = x.lnnx - n.∫ lnn - 1x dx

Integrals Involving sinh ax

$\int_{}^{}sinhaxdx=\frac{coshax}{a}$

$\int_{}^{}x.sinhaxdx=\frac{x.coshax}{a}-\frac{sinhax}{a^2}$

$\int_{}^{}x^2.sinhaxdx=\left(\frac{2}{a^3}+\frac{x^2}{a}\right)coshax-\frac{2x.sinhax}{a^2}$

$\int_{}^{}\frac{sinhaxdx}{x}=ax+\frac{(ax)^3}{3*3!}+\frac{(ax)^5}{5*5!}+...$

$\int_{}^{}\frac{sinhaxdx}{x^2}=\frac{-sinhax}{x}+a.\int_{}^{}\frac{coshaxdx}{x}$

$\int_{}^{}\frac{dx}{sinhax}=\frac{1}{a}\frac{ln(tan\frac{ax}{2})}{a}$

$\int_{}^{}sinh^2axdx=\frac{sinhax.coshax}{2a}-\frac{x}{2}$

$\int_{}^{}x.sinh^2axdx=\frac{x.sinh2ax}{4a}-\frac{cosh2ax}{8a^2}-\frac{x^2}{4}$

$\int_{}^{}\frac{dx}{sinh^2ax}=\frac{-cothax}{a}$

Integrals - 1 part
Integrals - 2 part
Integrals - 3 part
Integrals - 4 part

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