The definite integral has a unique value. A definite integral is denoted by ∫abf(x)dx, where a is called the lower limit of the integral and b is called the upper limit of the integral.
1.0Definite Integration Definition
A definite integral is denoted by ∫abf(x)dx which represents the algebraic area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis.
2.0Fundamental Theorem of Calculus
P–1 :If f is continuous on [a, b] then the function g defined by g(x) = ∫axf(t)dt, a ≤ x ≤ b is continuous on [a, b] and differentiable on (a, b) and g'(x) = f(x)
P–1 :If f is continuous on [a, b] then ∫abf(x)dx = F(b) – F(a) where F is any antiderivative of F, such that F' = f
3.0Properties of Definite Integrals
P–1 :∫abf(x)dx=∫abf(t)dt(change of variable does not change value of integral)
P–2 :∫abf(x)dx=−∫baf(x)dx
P–3 : ∫abf(x)=∫acf(x)dx+∫cbf(x)dx
P–4 :∫−aaf(x)dx=∫0a(f(x)+f(−x))dx=[02∫0af(x)dx if f(x) is odd if f(x) is even
P–5:∫abf(x)dx=∫abf(a+b−x)dx or ∫0af(x)dx=∫0af(a−x)dx (King)
P–6:∫02af(x)dx=∫0af(x)dx+∫0af(2a−x)dx=
[02∫0af(x)dx if f(2a−x)=−f(x) if f(2a−x)=f(x)(Queen)
4.0Walli’s Theorem
(a)∫0π/2sinnxdx=∫0π/2cosnxdx=n(n−2)…..(1 or 2)(n−1)(n−3)…..(1 or 2)K
where K={π/21 if n is even if n is odd
(b)∫0π/2sinnx⋅cosmxdx=(m+n)(m+n−2)(m+n−4)….1 or 2[(n−1)(n−3)(n−5)….1 or 2][(m−1)(m−3)….… or 2]K Where K={2π1 if both m and n are even (m,n∈N) otherwise
5.0Derivative of Antiderivatives(Newton-Leibnitz Theorem)
If F(x)=∫g(x)h(x)f(t)dt, then dxdF(x)=h′(x)f(h(x))−g′(x)f(g(x))
Proof: Let P(t)=∫f(t)dt⇒F(x)=∫g(x)h(x)f(t)dt=P(h(x))−P(g(x))