12
CLASS
Definite
Integration
Properties
SAMEER CHINCHOLIKAR
Sameer Chincholikar
IIT Roorkee
12+ Years of Teaching Experience
Trained Million+ students
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definite Integration:
Definition
Definite Integration : Definition
The numbers a and b are called the limits of integration, ‘a’ is called the
lower limit and ‘b’ the upper limit.
Evaluate :
Evaluate :
A. π B. 2π C. 3π D. None of these
Definite Integration using substitution
Evaluate :
A. B. C. -1 D. 1
Definite Integration using by parts
Evaluate :
Definite Integration and limits of integration
Evaluate :
Geometrical
understanding of
Definite Integration
Definite Integration : Geometrical understanding
Definite Integration : NOTE
Definite integration gives ‘effective’ or ‘net’ area under the curve.
Properties of
Definite Integration
Property-1: Dummy Variable
Property-2: Swapping the limits
Suppose f is such that f(-x) = -f(x) for every real x and
then
A. 10 B. 5 C. 0 D. -5
Property-3: Take a Break
In general:
If f(x) =
A. 0 B. 1 C. 2 D. 3
Evaluate :
A. 0 B. 1 C. 2 D. 3
, where[.] denotes the greatest integer function, equals
A. 2 + √2 B. 2 - √2 C. 1 + √2 D. √2 - 1
Evaluate :
Evaluate :
Evaluate :
Property-4:
Evaluate :
Evaluate :
Evaluate :
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Property-5:
Special Case:
Evaluate :
Evaluate :
NOTE :
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Evaluate :
Evaluate :
Property-6:
Evaluate :
Evaluate :
If f(sin x) dx, the value of k is
A. π/2 B. π C. π/3 D. 2π
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Evaluate :
Important Result:
Evaluate :
Evaluate :
Evaluate :
Property-7:
If f(x) is periodic function with period T then
1.
Property-7:
If f(x) is periodic function with period T then
2.
Property-7:
If f(x) is periodic function with period T then
3.
Property-7:
If f(x) is periodic function with period T then
4.
Property-7:
If f(x) is periodic function with period T then
5.
Evaluate :
Evaluate :
Evaluate :
Evaluate :
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