Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 9 Integration will enable students to study smartly.

## CHSE Odisha 12th Class Math Notes Chapter 9 Integration

Indefinite integral:

If \(\frac{d}{d x}\)F(x) = f(x) then the indefinite integral of f(x) w.r.t x is

∫f(x)dx = F(x) + C

which represents the entire class of anti-derivatives.

(a) Algebra of integrals:

(i) ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx

(ii) ∫af(x) dx = α ∫f(x) dx

(iii) ∫f(x) g(x) dx = f(x) ∫g(x) dx – ∫\(\left[(\frac{d}{d x} f(x)\right) \cdot \int g(x) d x]\) dx (Integration by parts)

(b) Some standard indefinite integrations:

(1) ∫x_{n }dx = \(\frac{x^{n+1}}{n+1}\) + C, n ≠ (-1)

(2) ∫\(\frac{d x}{x}\) = log_{e}|x| + C.

(3) ∫sin x dx = -cos x + C

(4) ∫cos x dx = sin x + C

(5) ∫tan x dx = -log |cos x| + C or log |(sec x)| + C

(6) ∫cot x dx = log |(sin x)| + C or -log |(cosec x)| + C

(7) ∫sec x dx = log |sec x + tan x| + C

(8) ∫cosec x dx = log |cosec x – cot x| + C

(9) ∫sec^{2 }x dx = tan x +C

(10) ∫cosec^{2} x dx = -cot x + C

(11) ∫sec x tan x dx = sec x + C

(12) ∫cosec x cot x dx = -cosec x + C

[Note: To integrate by parts choose 1st function according to I LATE]

Where I → Inverse trigonometric functions.

L → Logarithmic function

A → Algebraic function

T → Trigonometric function

E → Exponential function

(c) Techniques of integration:

Definite integration: