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Indefinite Integration

Indefinite Integration - Indefinite Integration Advanced...

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Indefinite Integration Advanced Level Pure Mathematics Advanced Level Pure Mathematics Calculus II Indefinite Integration 2 Method of substitution 3 Integration by Parts 6 Special Integration 10 Integration of Trigonometric Function 14 Reduction Formula 15 Prepared by Mr. K. F. Ngai Page 1 6 + = C x ln dx x 1

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Indefinite Integration Advanced Level Pure Mathematics INDEFINITE INTEGRATION Definition ) x ( f is said to be primitive function or anti-derivative of ) x ( g if ) x ( g ) x ( ' f = . Example x 2 ) x ( dx d 2 = 2 x is the primitive function of x 2 . Note Primitive function is not UNIQUE . Definition For any function ) x ( f if ) x ( F is the primitive function of ) x ( f , i.e. ) x ( f ) x ( ' F = , then we define the indefinite integral of ) x ( f w.r.t.x as c ) x ( F dx ) x ( f + = , where c is called the constant of integration . Theorem Two function ) x ( f and ) x ( h differ by a constant if and only if they have the same primitive function. Standard Results 1. + = c x ln dx x 1 2. c e dx e x x + = 3. + = c x sin xdx cos 4. + - = c x cos xdx sin 5. + = c x tan xdx sec 2 6. + - = c x cot xdx csc 2 7. + = c x sec xdx tan x sec 8. + - = c x csc xdx cot x csc 9. + = c a ln a dx a x x 10. c a a x x ln dx a x 1 2 2 2 2 + - + = - 11*. + = - - c a x sin dx x a 1 1 2 2 12*. c a x tan a 1 dx a x 1 1 2 2 + = + - 13. c a x a x ln dx a x 1 2 2 2 2 + + + = + Theorem (a) = dx ) x ( f k dx ) x ( kf (b) ± = ± dx ) x ( g dx ) x ( f dx ] ) x ( g ) x ( f [ . Prepared by Mr. K. F. Ngai Page 2
Indefinite Integration Advanced Level Pure Mathematics Example Prove c a ln a dx a x x + = proof Let x a y = . a ln x y ln = a ln dx dy y 1 = a ln y dx dy = dx dx dy = dx a ln y y = dx y a ln dx a x = c a ln a x + METHOD OF SUBSTITUTION Theorem ( CHANGE OF VARIABLE ) If ) t ( g x = is a differentiable function, = dt ) t ( ' g )) t ( g ( f dx ) x ( f .

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