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Page 2 of 6. INTEGRATION FORMULAE STANDARD INTEGRALS 1. ∫, ( )- ( ) , ( )- [where, n ≠ 1- 2. ∫ ( ) ( ) * ( )+ 3. ∫ ( ) ∫ ( ) [where, v be the function of x]
Integrals of Exponential and Logarithmic Functions. ∫ ln x dx = x ln x − x + C. + 1 x. + 1. x ∫ x ln xdx = ln x − + C. 2 + 1 ( n + 1 ) x dx = e x + C ∫.
A Review: The basic integration formulas summarise the forms of indefinite integrals for may of the functions we have studied so far, and the substitution method helps us use the table below to evaluate more complicated functions involving these basic ones.
Integration by parts: u dv = uv. −. v du + C. ax + b. Partial Fractions: to integrate a function like : (x + c)(x + d) ax + b A B A(x + d) + B(x + c) Write = + = , (x + c)(x + d) (x + c) (x + d) (x + c)(x + d) so ax + b = A(x + d) + B(x + c) = (A + B)x + (Ad + Bc), so a = A + B and b = Ad + Bc; solve for A and B .
Basic integration formulas. (x + c)(x + d) (x + c) (x + d) (x + c)(x + d) so ax + b = A(x + d) + B(x + c) = (A + B)x + (Ad + Bc), so a = A + B and b = Ad + Bc; solve for A and B . The approach for more general denomenator can be found in nearly any calculus textbook.
∫Integral Substitution: ( (𝑥))⋅ ′(𝑥) 𝑥=∫ ( ) , = (𝑥) Definite Integrals Rules: ∫Definite Integral Boundaries: (𝑥 ) 𝑥 =𝐹( )−𝐹( )=lim𝑥→ −𝐹𝑥− lim𝑥→ +𝐹(𝑥) )Odd Function: If (𝑥=− (−𝑥), then ∫ (𝑥) 𝑥 − =0
De nite (Riemann) Integral lim jj !0jj Xn i=1 f(c i) x i = Z b a f(x)dx Partitions of equal width: jj jj= x Partitions of unequal width: jj= max( x i) Continuity )Integrability The Converse is NOT True (see example below) Example: 2 0 bxcdx = 1 Area of a Region in a Plane Area of the region bounded by the graph of f, x-axis, and x = a and x = b ...
