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# Revision notes on Definite Integral

• If  then the equation f(x) = 0 has at least one root lying in (a, b) provided f is a continuous function in (a, b).

• If the function f is same then

• dx where c is any point lying inside or outside [a, b].

This holds true only when f is piecewise continuous in (a, b)

• if f(x) = -f(-x) i.e. f is an odd function

if f(x) = f(-x) i.e. f is an even function

• where f is a periodic function with period a

• where f(a+x) = f(x), i.e. a is the period of the function f

• If f(x) ≤ φ[x] for a ≤ x ≤ b then

• Gamma Function:

If n is a positive rational number, then the improper integral



is defined as the gamma function.

Γ(n+1) = n!

Γ1 = 1

Γ0 = ∞

Γ(1/2) = √π

### 

• If f(x) ≥ 0 on the interval [a, b] then

• ### Walli’s Formula

where K = , if both m and n are even (m, n ϵ N)

= 1 otherwise

• ### Leibnitz’s Rule:

If h(x) and g(x) are differentiable functions of x then

• For a monotonically increasing function in (a, b)

• Where f(x) is a continuous function on [a, b] and F’(x) = f(x).

• If f(x) = f(a – x), then

• If f(x) = – f (a – x), then

• If f(x) is a periodic function with period T, then

• In the above result, if n = 1, then

• The definite integral f(x)dx is in fact a limiting case of the summation of an infinite series, provided f(x) is continuous on [a, b] i.e.,

where h = b – a/n.

• The converse is also true i.e., if we have an infinite series of the above form, it can be expressed as definite integral.

• Method to express the infinite series as definite integral:

1. Express the given series in the form Σ 1/n f (r/n)

2. Then the limit is its sum when n → ∞, i.e. limn→∞ h Σ 1/n f(r/n)

3. Replace r/n by x and 1/n by dx and limn→∞ Σ by the sign of ∫

4. The lower and the upper limit integration are the limiting values of r/n for the first and the last terms of r respectively.

• Some particular case of the above are: