**Integration Formula In Pdf –**

**Formula Chart **

__Integration Formula On Standard Elementary Functions__

__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.

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, x^{p1/q1} ……x^{pk/qk} ). Substitute x = t^{m} is L.C.M. of q_{1} ,q_{2} …….q_{k}.

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

**# Integration of the form ∫R (x, √ax**^{2}** + bx + c) dx**

(i) Integrals of the form

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

M_{1 } , N_{1} , k are constants.

(ii) Integrals of the form I = ∫ [ P_{m} (x) / √ax^{2} + bx + c] dx

P_{m} (x) is a polynomial of degree m.

Where P_{m-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 – a_{1} )^{m} √ax^{2} + bx + c

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

**Note :** For the integration of the form ∫ √ax^{2} + bx + c dx or ∫ dx/ √ax^{2} + bx + c or ∫ dx /ax^{2} + bx + c.

First convert ax^{2} + bx + c in the form A^{2} + X^{2} , X^{2} – A^{2} , or A^{2} – X^{2} where A is constant and x is px + q.

**Following results is used**

**# Integration of the form of ∫ x ^{m} ( a + bx^{n}) 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 = t^{k} where k is L.C.M. of denominator of m and n.

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

Case 4. \( \frac {m+1}{n} + P \) is an integer, put a + bx^{n} = t^{α} x^{n} where α is denominator of fraction P.

__Download Integration Formula-I In Pdf click here __

__Download Integration Formula-I In Pdf click here__

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