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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:

Important Results:





















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