Definate Integration
Leibnitz's Rule
Approximate Formulas for Definite Integrals
Solved Examples
Properties of Definate Integral
In the following the interval from x = a to x = b is subdivided into n equal
parts by the points a = x
0, x
2, . . ., x
n-1, x
n = b and we let y
0 = f(x
0),
y
1 = f(x
1), y
2 = f(x
2), . . ., y
n = f(x
n), h = (b - a)/n.
Rectangular formula
Trapezoidal formula
Simpson’s formula (or parabolic formula) for n even
Example : Evaluate
Solution : Putting e
x = t
e
xdx = dt
When x = 0, t = e
0 = 1
When x = 1, t = e
1 = e
= tan
-1e - tan
-11
= tan
-1e - π/4
Example : Evaluate
Solution : Putting x = tanθ
dx = sec
2θdθ
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 x
2 = t
2xdx = dt
xdx = ½dt
When x = 0, t = 0
And, when x = 1, t = 1
