List of Integration Formulas | Basic ,Trig, Substitution, Parts, Definite | Class 12 Maths

Here is the Integration Formulas List.  The formula list is divided into below sections

a. Basic integration formulas

b.Integration formulas for Trigonometric Functions

c. Integration formulas Related to Inverse Trigonometric Functions

d. Algebra of integration

e. Integration by Substitution

f. Special Integrals Formula

g. Integration by Parts

h. Some special Integration Formulas derived using Parts method

i. Integration of Rational algebraic functions using Partial Fractions

j. Definite Integrals

k. Properties of Definite Integrals

l. Integration as Limit of Sum

Basic Integration formulas

$\int (c) = x + C$  ( Where c is a constant)

$\int (cx) = \frac {cx^2}{2} + C$ ( Where c is a constant)

$\int (x^n) = \frac {x^{n+1}}{n+1}$

$\int (e^x) = e^x + C$

$\int (\frac {1}{x}) = ln |x| + c$

$\int (a^x) = \frac {a^x}{ log a} + C$

$\int (log_{a} x) =\frac {1}{x ln a} + C$

Integration formulas for Trigonometric Functions

$\int (\cos x) = \sin x + C$

$\int (\sin x) = – \cos x + C$

$\int ( \sec^2 x) = \tan x + C$

$\int (\csc^2 x) = -\cot x + C$

$\int ( \sec (x) \tan (x) )=\sec x + C$

$\int (\csc (x) \cot (x)) = -\csc x + C$

$ \int (\tan x) = ln |\sec x| + C$

$ \int (\cot x) = ln |\sin x| + C$

$ \int (\sec x) = ln |\sec x + \tan x| + C$

$ \int (\csc x) = ln |\csc x – \cot x| + C$

Integration formulas Related to Inverse Trigonometric Functions

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

$\int (\frac {1}{\sqrt {1-x^2}}) = – \cos ^{-1}x  +C$

$\int ( \frac {1}{1 + x^2}) =\tan ^{-1}x + C$

$\int ( \frac {1}{1 + x^2}) = -\cot ^{-1}x + C$

$\int (\frac {1}{|x|\sqrt {x^-1}}) = -sec^{-1} x + C $

$\int (\frac {1}{|x|\sqrt {x^-1}}) = -cosec^{-1} x + C $

Algebra of integration

Multiplication by Constant

$\int [cf(x)] dx = c \int f(x) dx $

Example

$\int [2x ] dx = 2 \int (x) dx =x^2 + C$

Addition and Subtraction

$\int [f(x)+g(x)] dx=\int f(x)  dx+ \int g(x) dx$

$\int [f(x)-g(x)]dx=\int f(x) dx – \int  g(x) dx$

Example

$\int [sinx -cos x ] dx = \int sin x  dx- \int cos x dx=-cos (x) – sin(x) + C$

Integration by Substitution

A. if $ \int f(x) dx =  g(x) $ then $\int f(ax+ b) = \frac {1}{a} g(x) $

$\int (ax+b)^n = \frac {1}{a} \frac {(ax+ b)^{n+1}}{n+1} + C$

$\int e^{ax+b} =\frac {1}{a}  e^{ax+b} + C$

$\int (\frac {1}{ax+b}) = \frac {1}{a} ln |ax +b| + c$

$\int a^{bx+c} = \frac {1}{b} \frac {a^{bx+c}}{ log a} + C$

$\int \cos (ax+b) = \frac {1}{a}  \sin (ax+b) + C$

$\int \sin (ax+b) = – \frac {1}{a} \cos (ax+b) + C$

$\int \sec^2 (ax+b) = \frac {1}{a}  \tan (ax +b) + C$

$\int \csc^2 (ax+b) = – \frac {1}{a}  \cot (ax+b)+ C$

$ \int \tan (ax+b) =- \frac {1}{a}  ln |\cos (ax+b)| + C$

$ \int \cot (ax+b) = \frac {1}{a}  ln |\sin (ax+b)| + C$

$ \int \sec (ax+b) =\frac {1}{a} ln |\sec (ax+b) + \tan (ax+b)| + C$

$ \int \csc (ax+b) = \frac {1}{a} ln |\csc (ax+b) – \cot (ax+b)| + C$

B.  $\int \frac {f^{‘} (x)}{f(x)} dx  = ln | f(x)| + C$

Example

$\int \frac {1}{1 + e^{-x}} dx = \int \frac {1}{1 + 1/e^x} dx = \int \frac {e^x}{1+ e^x} dx$

Now this above form

$= ln |1 + e^x| + C$

C.  $\int  [f(x)]^n f^{‘} x dx = \frac { [f(x)]^{n+1}}{n +1 }  , n \ne -1 $

Special Integrals Formula

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

$\int \frac {1}{x^2 – a^2} dx = \frac {1}{2a} ln  |\frac {x-a}{x+a}| + C$

$\int \frac {1}{a^2 – x^2} dx = \frac {1}{2a} ln  |\frac {a+x}{a-x}| + C$

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

$\int \frac {1}{\sqrt {a^2  + x^2}} dx =  ln |x + \sqrt {a^2  + x^2}|   + C$

$\int \frac {1}{\sqrt {x^2  – a^2}} dx =  ln |x + \sqrt {x^2  – a^2}|   + C$

Integration by Parts

A. $\int f(x) g(x) dx= f(x)  (\int g(x) dx ) – \int \left \{ \frac {df(x)}{dx} \int g(x) dx \right \}dx$

We can decide first function using the word ILATE

I -> Inverse trigonometric functions

L -> Logarithmic functions

A-> Algebraic functions

T -> trigonometric functions

E -> Exponential functions

B. $\int e^x{ f(x) + f^{‘} (x) } dx =  e^x f(x)  + C$

Some special Integration Formulas derived using Parts method

$ \int \sqrt {a^2 – x^2} dx = \frac {1}{2} x \sqrt {a^2 – x^2} + \frac {1}{2} a^2 \sin^{-1} \frac {x}{a} + C$

$ \int \sqrt {a^2 + x^2} dx = \frac {1}{2} x \sqrt {a^2 + x^2} + \frac {1}{2} a^2  ln |x +\sqrt {a^2 + x^2}|  + C$

$ \int \sqrt {x^2 -a ^2} dx = \frac {1}{2} x \sqrt {x^2 – a^2} – \frac {1}{2} a^2  ln |x +\sqrt {x^2 – a^2}|  + C$

The above formula can be to use to integrate the below type of function

$ \int \sqrt {ax^2 + bx + c} dx$

We can convert $ax^2 + bx + c$ into above using square method

$ \int  (px +q) \sqrt {ax^2 + bx + c} dx$

We can express $px + q = \lambda \frac {d}{dx} (ax^2 + bx +c) + \mu$

We find the values of $ \lambda$ and $\mu$

Now this will get converted into entities. One of the integration can be obtained from above formula and one from

$\int  [f(x)]^n f^{‘} x dx = \frac { [f(x)]^{n+1}}{n +1 }  , n \ne -1 $

Integration of Rational algebraic functions using Partial Fractions

$ \int \frac {px +q}{(x-a)(x-b)} dx =\int  \left \{  \frac {A}{x-a} + \frac {B}{x-b}   \right \} dx$

$ \int \frac {px^2 +qx + r}{(x-a)(x-b)(x-c)} dx =\int  \left \{  \frac {A}{x-a} + \frac {B}{x-b}  + \frac {C}{x-c}  \right \} dx$

$ \int \frac {px +q}{(x-a)^2} dx =\int  \left \{  \frac {A}{x-a} + \frac {B}{(x-a)^2}   \right \} dx$

$ \int \frac {px^2 +qx + r}{(x-a)^2(x-c)} dx =\int  \left \{  \frac {A}{x-a} + \frac {B}{(x-a)^2}  + \frac {C}{x-c}  \right \} dx$

$ \int \frac {px^2 +q+r}{(x-a)(x^2 + bx +c)} dx =\int  \left \{  \frac {A}{x-a} + \frac {Bx +C}{x^2 + bx +c}   \right \} dx$

where $x^2 + bx +c$ is a irreducible quadratic

Definite Integrals

if  $\int f(x)  dx= g(x)$
$\int_{a}^{b} f(x) dx =g(b) -g(a)$

Properties of Definite Integrals

$\int_{a}^{b} f(x) dx= \int_{a}^{b} f(t) dt $

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

$\int_{a}^{b} f(x) dx= \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx  $

where a<c <b

if (x) is a continuous function defined on [0,a],then

$\int_{0}^{a} f(x) dx=\int_{0}^{a} f((a-x)) dx$

$\int_{-a}^{a} f(x) dx=\int_{0}^{a} f((a-x)) dx$

$\int_{-a}^{a} f(x) dx= \begin{cases}
2 \int_{0}^{a} f(x) dx & ,  f(x) =f(-x) \\
0 &, f(x) =-f(x)
\end{cases} $

Integration as Limit of Sum

$\int_{a}^{b} f(x) dx = \lim_{h \rightarrow 0} h[f(a) + f(a+h) + f(a+2h)+…..+f(a + (n-1)h)]$

where $h= \frac {b-a}{n}$

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