Summary Definite Integral Made Easy, ISBN: 9788131802069 Engineering maths

Definite Integrals
Let f ( x ) be a function defined on the interval [a , b] and F ( x ) be its
b
anti-derivative. Then, ∫ f ( x ) dx = F ( b) − F ( a ) is defined as the
a
definite integral of f ( x ) from x = a to x = b.
The numbers a and b are called upper and lower limits of integration,
respectively.

Fundamental Theorem of Calculus
There is a connection between indefinite integral and definite integral
is known as fundamental theorem of calculus.
First Fundamental Theorem
Let f be a continuous function defined on the closed interval [a , b] and
x
let A( x ) be the area of function i.e. A( x ) = ∫ f ( x )dx. Then, A′ ( x ) = f ( x )
a
for all x ∈ [a , b].
Second Fundamental Theorem
Let f be a continuous function defined on the closed integral [a , b] and
F be an anti-derivative of f. Then,
b
∫a f ( x )dx = [F( x )]a = F( b) – F( a ).
b



Evaluation of Definite Integrals by Substitution
Consider a definite integral of the following form
b
∫ a f [ g( x )] ⋅ g ′ ( x ) dx
Step I Substitute g( x ) = t ⇒ g ′ ( x ) dx = dt
Step II Find the limits of integration in new system of variable
i. e. , the lower limit is g( a ) and the upper limit is g( b) and
g ( b)
the new integral will be ∫ g ( a ) f ( t) dt.
Step III Evaluate the integral, so obtained by usual method.

, Properties of Definite Integral
b b
1. ∫ a f ( x ) dx = ∫ a f ( t) dt
b a
2. ∫ a f ( x ) dx = − ∫ b f ( x ) dx
a
3. ∫ a f ( x ) dx = 0
b c b
4. ∫ a f ( x ) dx = ∫ a f ( x ) dx + ∫ c f ( x ) dx, where a < c < b
Generalisation
If a < c1 < c2 < K < cn − 1 < cn < b, then
b c1 c2 c3
∫ a f ( x ) dx = ∫ a f ( x ) dx + ∫
c1
f ( x ) dx + ∫c 2
f ( x ) dx

cn b
+K + ∫c n −1
f ( x ) dx + ∫c
n
f ( x ) dx
a a
5. ∫ 0 f ( x ) dx = ∫ 0 f ( a − x ) dx
a f(x) a
Deduction ∫ dx =
0 f(x) + f(a − x) 2
b b
6. ∫ a f ( x ) dx = ∫ a f ( a + b − x ) dx
b f(x) b− a
Deduction ∫ a f ( x ) + f ( a + b − x ) dx = 2
2a a a
7. ∫0 f ( x ) dx = ∫ 0 f ( x ) dx + ∫ 0 f ( 2a − x ) dx
a a a
8. ∫ − a f ( x ) dx = ∫ 0 f ( x ) dx + ∫ 0 f ( − x ) dx
 2 a f ( x ) dx if, f ( 2a − x ) = f ( x )
2a  ∫
9. ∫0 f ( x ) dx =  0
 0, if f ( 2a − x ) = − f ( x )

 0, if f ( a + x ) = − f ( b − x )
b 
10. ∫a f ( x ) dx =  a + b
 2 2 f ( x ) dx , if f ( a + x ) = f ( b − x )
 ∫a
 2 a f ( x ) dx , if f ( x ) is even i. e. f ( − x ) = f ( x )
a  ∫
11. ∫− a f ( x ) dx =  0
 0, if f ( x ) is odd i. e. f ( − x ) = − f ( x )
