Indefinite Integration Formulas

Basic Concepts & Indefinite Integration Formulas

• It is a process of finding the limit of a sum of a certain number of elements, as the number of elements tends to infinity when each of the elements tends to zero.
• Integration is a reverse process of differentiation.

If {d (F(x) + c)/dx} = f(x), then

Integration of f(x).dx = F(x) + C ; Here ‘C’ is constant of integration

F(x) is Integral of f(x) or Anti-Derivative of f(x) or Primitive of f(x)

• In differentiation dy/dx is used as a composite symbol or as an operator. In ‘Methods of Differentiation’ these do not have separate meanings.
• But in Integration, it is treated as a fraction where ‘dy’ is called differential of ‘y’ and ‘dx’ is called differential of ‘x’.
• For a single variable function y=f(x), differentiation of ‘y’ is denoted as ‘dy’

i.e., dy = Product of derivative of F(x) w.r.t ‘x’ and differential of x is denoted by ‘dx’.

Ex-  y = cos x then dy = -sin x dx

Geometrical Interpretation

y = ∫ f(x) dx denotes a family of curves such that derivative at x= x₁ of every member of the family is same i.e. tangent

drawn to each member of the family at x = x₁ is parallel.

Ex- y = ∫ 2x dx = x² + C

NOTE-

• Anti-derivative of a continuous function is always differentiable.
• Anti-derivative of a periodic function needn’t be a periodic function.

Differentiation Notes

Integral of Standard Functions

∫ (xⁿ)dx = (xⁿ⁺¹)/(n+1) + C ; n ≠ -1

∫ (ax+b)ⁿ dx = (ax+b)ⁿ⁺¹)/a(n+1) + C ; n ≠ -1

∫1/x dx = ln |x| + C

∫ 1/(ax+b) dx = 1/a ln |ax+b| +C

∫ a^x dx = a^x/ln a + C ; a>0

∫ a^px+q dx = a^px+q/p ln a + C ; a>0

∫ e^x dx = e^x + C

∫ e^px+q dx = (e^px+q)/a  + C

Trigonometric Integrals

∫ sin x dx = – cos x +C

∫ sin (ax+b )dx = – 1/a cos (ax+b)  + C

∫ cos x dx =  sin x +C

∫ cos (ax+b) dx = 1/a sin (ax+b)  + C

∫ sec² x dx =  tan x +C

∫ sec² (ax+b) dx = 1/a tan (ax+b) + C

∫ cosec² x dx =  – cot x + C

∫ cosec² (ax+b) dx = – 1/a cot (ax+b) + C

∫sec x tan x dx = sec x + C

∫sec (ax+b) tan (ax+b) dx = 1/a sec (ax+b) + C

∫ cosec x cot x dx = – cosec x + C

∫ cosec (ax+b) cot (ax+b) dx = – 1/a cosec (ax+b) + C

Inverse Trigonometric Integrals

∫dx/ (1+x²) dx = tan^-1 x + C

∫dx/ (a² + x²) dx = 1/a tan^-1 x + C

∫dx/ (1- x²)^1/2 dx = sin^-1 x + C

∫dx/ (a² – x²)^1/2 dx = 1/a sin^-1 (x/a) + C

∫dx/ x( x² – 1)^1/2 dx = sec^-1 x + C

∫dx/ x( x² – a²)^1/2 dx = 1/a sec^-1 (x/a) + C