Integration Formula-I In Pdf

Formula Chart

Integration Formula On Standard Elementary Functions



# Integration by substitution

Basic rule of substitution

∫ f(x) dx = ∫ f {Φ(z)·Φ (z)} dz                     ;   Substitute   x =Φ(z)

# Integration by parts

If u and v be two functions of x then  ∫ uv dx = u ∫ v dx – ​\( \int (\frac{du}{dx}\int v dx)dx \)​ is

known as formula for integration by parts.

Note : In finding Integrals by this method proper choice of u and v is essential. In general the funcyion as u is taken which comes first in the world ILATE.

Where,                   I  – Inverse circulated function

                                L  – Logarithmic function

                                A  – Algebraic function

                                T  – Trigonometric function

                                E  – Exponential function

Note :

*  if both the function s are trigonometric, take that function as v whose integral is simpler.

** If both functions are algebraic take that the function as ‘u’ whose d.c is simpler.

# Integration of rational function :

If integrand is of the P(x)/Q(x), where P(x) and Q(x) are polynomials of x.

Case I. If P(x)/Q(x) is proper rational function ( i.e. highest power of x in Q(x) > P(x) integration by partial function is done.

Case II. If P(x)/Q(x) is improper rational function (i.e. highest power of x in Q(x) < P(x)). Then it is made proper rational function by simple division and then partial fraction is used.

integration formula of rational function

f(x) is integrated by general rule and h(x)/Q(x) is integrated by partial fraction.

# Integration of certain irrational expressions :

(i) If the integrand is a rational function of a fractional powers of an independent variable x i.e. the function R (x, xp1/q1  ……xpk/qk ). Substitute x = tm  is L.C.M. of q1  ,q2  …….qk.

(ii) If the integrand is rational function x and fractional powers of a linear fractional function of the form

integration formula of irrational function

# Integration of the form ∫R (x, √ax2 + bx + c) dx

(i) Integrals of the form

integration formula

Substitute ​\( x + \frac {b}{2a}=t \)​ and reduce it to the form

integration formula notes

M1 , N1 , k are constants.

(ii) Integrals of the form I = ∫ [ Pm (x) / √ax2 + bx + c] dx

Pm (x) is a polynomial of degree m.

integration of special function

Where Pm-1 (x) is polynomial of degree (m-1) and k is constant. Coefficient of  is determined by the method of undetermined coefficients.

(iii) Integrals of the form ∫ dx / ( x – a1 )m √ax2 + bx + c

Substitute ( x – a1 ) = ​\( \frac {1}{t} \)

Note : For the integration of the form ∫ √ax2 + bx + c dx or  ∫ dx/ √ax2 + bx + c or ∫ dx /ax2 + bx + c.

First convert ax2 + bx + c in the form A2 + X2 , X2 – A2 , or A2 – X2 where A is constant and x is px + q.

Following results is used

integration formula of algebraic formula

# Integration of the form of  ∫ xm ( a + bxn) P dx ; m, n, p are rational number

Case 1. If P is a positive integer then integrand is expanded by binomial expansion.

Case 2. If P is negative integer substitute x = tk where k is L.C.M. of denominator of m and n.

Case 3. ​\( \frac {m+1}{n} \)​  is an integer, put a + bxn = tα  where α is denominator of fraction P.

Case 4. ​\( \frac {m+1}{n} + P \)​ is an integer, put a + bxn = tα xn  where α is denominator of fraction P.

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