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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:  ### Course Features

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