Indefinite Integration – Direct Subtitution
Definition: If f(x) and g(x) are two functions such that f'(x) = g(x) then f(x) is called antiderivative or primitive of g(x) with respect to x
Note1: If f(x) is an antiderivative of g(x) then f(x)+c is also an antiderivative of g(x) for all c ϵ wR
If F(x) is an antiderivative of f(x) then F(x)+c, c ϵR is called indefinite integration of f(x) with respect to x.it is denoted by ∫ f(x) dx .the real number is called constant of integration
Direct Substitution: If the integral is of the form ∫ f(g(x)) g'(x) dx, then put g(x) = t, provided ∫ f(t) exists.
1. ∫ f ‘(x)/f(x) dx = ln|f(x)|+c
The given integral is ∫ f'(x)/f(x) dx
Put f(x) = t
⇒ f'(x) dx = dt
⇒ ∫ dt/t = ln |t| + c
= ln |f (x)| + c.
2. ∫ f ‘(x)/√f(x) dx = 2 √f(x) + c
The given integral is ∫ f'(x)/√f(x) dx
Put f (x) = t
Then the given integral becomes
= ∫ dt/√t = 2 √t + c
= 2 √f(x) + c.
Example: Evaluate ∫ (2x² + 1) sin² (2x³ + 3x) dx.
Solution: Let z = 2x³ + 3x
dz = (6x² +3)dx
dz = 3( 2x² +1)dx
= 1/3∫ sin2z dz
= 1/3 ∫ (1–cos2z)/2 dz
= 1/6 ∫ (1–cos 2z)dz
= 1/6 [z – sin2z / 2] + c
= 1/6 [2x³ + 3x – sin(4x³ + 6x)/2] + c
Hence ∫ (2x² + 1) sin² (2x³ + 3x) dx = 1/6 [2x³ + 3x – sin(4x³ + 6x)/2] + c.