Mathematics- Class XII

Unit III - Calculus

Chapter - Integrals

Topic – Definite integrals

Subtopic – Properties of definite integrals ,P0,P1,P2(PART-A)

Outline

• Definitions

• Proofs of Properties P₀ ,P_1, P₂

• Examples with solutions

• Assignment

P₀ :If f(x) is continuous function in the closed interval [a,b] then

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• The value of the Definite integral does not depend on the variable w.r.t integration is carried out.

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PROOF

• Suppose = F(x) +c

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then L.H.S =

If = F(x) +c then

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R.H.S = = F(b)-F(a)

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therefore LHS = RHS. i.e

dx

dx

= F(t) +C

PROPERTY:- P1

• If we interchange the limits of the value of the definite integral changes its sign

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• If lower and upper limits of definite integral are

equal then value of the definite integral is zero

Proof( P₁ )

Let F be the anti derivation of ‘f’.

By second fundamental theorem of calculus,

we have

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Here, we observe that, if a=b, then

PROPERTY-2

• F(x) is continuous on closed interval [a,b]

and then

a < c < b

_PROOF

• Let F(x) be a primitive of f(x) on [a,b].

then by fundamental theorem of definite integral

………..(1)

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…...........(2)

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……………(3)

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• Adding (2) and (3)

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= F(c)-F(a) + F(b) – F(c).

= F(b) – F(a).

Therefore

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ASSIGNMENT