Definate Integration

Leibnitz's Rule
Approximate Formulas for Definite Integrals
Solved Examples

Properties of Definate Integral




  1. , where




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Leibnitz's Rule







Approximate Formulas for Definite Integrals

In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = x0, x2, . . ., xn-1, xn = b and we let y0 = f(x0), y1 = f(x1), y2 = f(x2), . . ., yn = f(xn), h = (b - a)/n.

Rectangular formula


Trapezoidal formula






Simpson’s formula (or parabolic formula) for n even







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Solved Examples


Example
: Evaluate


Solution : Putting ex = t
                exdx = dt

When x = 0, t = e0 = 1
When x = 1, t = e1 = e






= tan-1e - tan-11
= tan-1e - π/4

Example
: Evaluate


Solution : Putting x = tanθ
                      dx = sec2θdθ

When x = 0, θ = 0
When x = 1, θ = π/4





















Now futher, Integrate the integral by parts taking θ as the first and sinθ as the second function.

Example
: Evaluate


Solution : Let


                  Let x2 = t
                  2xdx = dt
                  xdx = ½dt

When x = 0, t = 0
And, when x = 1, t = 1

Example
: Evaluate



Solution
:  



                                                                                 Using Property 5

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