Press "Enter" to skip to content

Integral formulas: Indefinite integral, Definite integral

integral formulas

Indefinite integral formulas:

Basic integral formulas:

  1. \(\large \int  \; dx = x + C\)
  2. \(\large \int a \; dx = ax + C\)
  3. \(\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1\)
  4. \(\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1\)

Integral formulas of trigonometric functions:

∫ sin x dx = -cos x + c

∫ cos x dx = sin x + c

∫ tan x dx = log|sec x| + c

∫ cot x dx = log|sin x| + c

∫ sec x dx = log|sec x + tan x| + c

∫ cosec x dx = log|cosec x – cot x| + c

∫ sec2x dx = tan x + c

∫ cosec2x dx = -cot x + c

∫ sec x tan x dx = sec x + c

∫ cosec x cot x dx = – cosec x + c

Integral formulas for inverse trigonometric functions:

 \(\large \int \frac{1}{\sqrt{1-x^{2}}} \; dx = \sin^{-1} x + C\)

\(\large \int \frac{1}{\sqrt{1+x^{2}}} \; dx = \tan^{-1} x + C\)

\(\large \int \frac{1}{|x|\sqrt{x^{2}-1}} \; dx = \sec^{-1} x + C\)

\(\large \int \frac{dx}{\sqrt{x^{2}+ {a}^2}} \; dx = \frac{1}{a}\;{tan}^{-1} \frac{x}{a} + C\)

\(\large \int \frac{dx}{\sqrt{a^{2}- {x}^2}} \; dx = {sin}^{-1} \frac{x}{a} + C\)

∫ sin-1x dx = x sin-1x + \(\sqrt{1-{x}^2}\) + c

∫ cos-1x dx = x cos-1x – \(\sqrt{1-{x}^2}\) + c

∫ tan-1x dx = x tan-1x – ln\(\sqrt{1 + {x}^2}\) + c

∫ cot-1x dx = x cot-1x +ln\(\sqrt{1 + {x}^2}\) + c

∫ sec-1x dx = x sec-1x – ln|x + \(\sqrt{|{x}^2-1|}\)| + c

∫ cosec-1x dx = x cosec-1x + ln|x + \(\sqrt{|{x}^2-1|}\)| + c

Integral formulas for exponential trigonometric functions:

\(\large \int \sin^{n} (x) dx = \frac{-1}{n} \sin^{n-1} (x) \cos (x) +\frac{n-1}{n} \int \sin^{n-2} (x) dx\)

\(\large \int \cos^{n} (x) dx = \frac{1}{n} \cos^{n-1} (x) \sin (x) +\frac{n-1}{n} \int \cos^{n-2} (x) dx\)

\(\large \int \tan^{n} (x) dx = \frac{1}{n-1} \tan^{n-1} (x) +\int \tan^{n-2} (x) dx\)

\(\large \int \sec^{n} (x) dx = \frac{1}{n-1} \sec^{n-2} (x) \tan (x) + \frac{n-2}{n-1} \int \sec^{n-2} (x) dx\)

\(\large \int \csc^{n} (x) dx = \frac{-1}{n-1} \csc^{n-2} (x) \cot (x) + \frac{n-2}{n-1} \int \csc^{n-2} (x) dx\)

∫ sin2x dx = \(\frac{1}{2}\)x – \(\frac{1}{4}\)sin 2x + c

∫ cos2x dx = \(\frac{1}{2}\)x + \(\frac{1}{4}\)sin 2x + c

∫ tan2x dx = tan x – x + c

∫ cot2x dx = -cot x – x + c

Integral formulas for logarithmic functions:

∫ ax dx = \(\frac{{a}^x}{log\:a}\) + c

∫ \(\frac{1}{x}\) dx = log|x| + c

∫ \(\frac{dx}{{x}^2 –{a}^2}\) =\(\frac{1}{2a}\) log|\(\frac{x\:-\:a}{x\:+\:a}\)| + c

∫\(\frac{dx}{{a}^2 –{x}^2}\) = \(\frac{1}{2a}\) log|\(\frac{a\:+\:x}{a\:-\:x}\)| + c

∫\(\frac{dx}{\sqrt{{x}^2 –{a}^2}}\) = log |x + \(\sqrt{{x}^2 – {a}^2}\)| + c

∫\(\frac{dx}{\sqrt{{x}^2 +{a}^2}}\) = log |x + \(\sqrt{{x}^2 + {a}^2}\)| + c

Integral formulas for partial fractions:

\(\frac{px\:+\:q}{(x-a)(x-b)}\) = \(\frac{A}{x\:-\:a} + \frac{B}{x\:-\:b}\) , a ≠ b

\(\frac{px\:+\:q}{(x-a)^{2}}\) = \(\frac{A}{x\:-\:a} + \frac{B}{(x-a)^{2}}\)

\(\frac{p{x}^2+qx+r}{(x-a)(x-b)(x-c)}\) = \(\frac{A}{x\:-\:a} + \frac{B}{x\:-\:b} + \frac{C}{x\:-\:c}\)

\(\frac{p{x}^2+qx+r}{(x-a)^{2}(x-b)}\) = \(\frac{A}{x\:-\:a} + \frac{B}{(x-a)^{2}} + \frac{C}{x-b}\)

\(\frac{p{x}^2+qx+r}{(x-a) ({x}^2 + bx +c)}\) = \(\frac{A}{x-a} + {Bx+C}{{x}^2 + bx +c}\)

Where x2 +bx +c can’t be factorized further.

Integral Formulas by parts:

$$\large \int u\;v\;dx=u\int v\;dx-\int\left(\frac{du}{dx}\int v\;dx\right)dx$$

In this formula there is simple rule to know first and 2nd function i.e. u and v:

I – Inverse Function

L – Logarithmic Function

A – Algebraic Function

T- Trigonometric Function

E- Exponential Function

∫ ex [f(x) + f’(x)] dx = ∫ex f(x) dx + c

Some Special integral formulas:

∫ \(\sqrt{{x}^2 – {a}^2}\) dx = \(\frac{x}{2}\;\sqrt{{x}^2 – {a}^2}\) – \(\frac{{a}^2}{2}\)log|x + \(\sqrt{{x}^2 – {a}^2}\)| + c

∫\(\sqrt{{x}^2 + {a}^2}\) dx = \(\frac{x}{2}\;\sqrt{{x}^2 + {a}^2}\) + \(\frac{{a}^2}{2}\)log|x + \(\sqrt{{x}^2 + {a}^2}\)| + c

∫\(\sqrt{{a}^2 – {x}^2}\) dx = \(\frac{x}{2}\;\sqrt{{x}^2 – {a}^2}\) + \(\frac{{a}^2}{2}\) sin-1\(\frac{x}{a}\) + c

Definite integral formulas:

Actual Rule:

$$\large \int_{a}^{\infty}f(x)dx=\lim_{b\rightarrow \infty}\left [ \int_{a}^{b}f(x)dx\right ]$$

Or we can write:

$$\large \int_{a}^{b}f(x)dx=F(b)-F(a)$$

Here a is lower limit and b, ∞ are upper limits.

Some rational formula for definite integration:

\(\large \int_{a}^{\infty }\frac{dx}{x^{2}+a^{2}}=\frac{\pi }{2a}\)

\(\large \int_{a}^{\infty }\frac{x^{m}dx}{x^{n}+a^{n}}=\frac{\pi a^{m-n+1}}{n\sin \left ( \frac{(m+1)\pi }{n} \right )},0< m+1< n\)\)

\(\large \int_{a}^{\infty }\frac{x^{p-1}dx}{1+x}=\frac{\pi }{\sin (p\pi )},0< p< 1\)

\(\large \int_{a}^{\infty }\frac{x^{m}dx}{1+2x\cos \beta +x^{2}}=\frac{\pi \sin (m\beta )}{\sin (m\pi )\sin \beta }\)

\(\large \int_{a}^{\infty }\frac{dx}{\sqrt{a^{2}-x^{2}}}=\frac{\pi }{2}\)

\(\large \int_{a}^{\infty }\sqrt{a^{2}-x^{2}}dx=\frac{\pi a^{2}}{4}\)

Definite integration formula for trigonometric functions:

\(\large \int_{0}^{\pi }\sin(mx)\sin (nx)dx=\left\{\begin{matrix} 0 & if\;m\neq n\\ \frac{\pi }{2} & if\;m=n \end{matrix}\right.\;m,n\;positive\;integers\)

\(\large \int_{0}^{\pi }\cos (mx)\cos (nx)dx=\left\{\begin{matrix} 0 & if\;m\neq n\\ \frac{\pi }{2} & if\;m=n \end{matrix}\right.\;m,n\;positive\;integers\)

\(\large \int_{0}^{\pi }\sin (mx)\cos (nx)dx=\left\{\begin{matrix} 0 & if\;m+n\;even\\ \frac{2m}{m^{2}-n^{2}} & if\;m+n\;odd \end{matrix}\right.\;m,n\;integers\)

 

More from Calculus
Relation and Functions Limits Formula
Continuity Rules Differentiability Rules
Derivative Formula Inverse Trigonometric function
Formulas
Application of Integrals Logarithm Formulas

 

Share with your Friends

Be First to Comment

Leave a Reply

error: Content is protected !!