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Piecewise Definite integration - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Piecewise Definite integration is considered one the most difficult concept.

  • 43 Questions around this concept.

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The value of ∫0π|cos⁡x|3dx is :

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The value of the integral

∫−π/2π/2sin2⁡x1+exdx

is

 

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Concepts Covered - 1

Piecewise Definite integration

Property 5

∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx where c∈R

This property is useful when the function is in the form of piecewise or discontinuous or non-differentiable at x=c in (a,b).
Let
ddx(F(x))=f(x)∫acf(x)dx+∫cbf(x)dx=F(x)|ac+F(x)|cb=F(c)−F(a)+F(b)−F(c)=F(b)−F(a)=∫abf(x)dx
∴∫acf(x)dx+∫cbf(x)dx

The above property can also be generalized into the following form

∫abf(x)dx=∫ac1f(x)dx+∫c1c2f(x)dx+…+∫cnbf(x)dx
where, a<c1<c2<…<cn−1<cn<b.

Property 6

∫0af(x)dx=∫0a/2f(x)dx+∫0a/2f(a−x)dx

Proof:
From the previous property,
∫0af(x)dx=∫0a/2f(x)dx+∫a/2af(x)dx

Put x=a−t⇒dx=−dt in the second integral, when x=a/2, then t=a/2 and when x=a, then t=0
∴∫0af(x)dx=∫0a/2f(x)dx+∫a/20f(a−t)(−dt)=∫0a/2f(x)dx+∫0a/2f(a−t)dt∫0af(x)dx=∫0a/2f(x)dx+∫0a/2f(a−x)dx

 

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Piecewise Definite integration

Study other Related Concepts

Piecewise Definite integration Current Topic

Integration as an Inverse Process of Differentiation
Integration of Trigonometric Functions
Integration by Substitution
Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions)
Integrals of Particular Function
Trigonometric Integrals
Integration By Parts Formula
Integration Using Partial Fraction
Integration of Irrational Algebraic Function
Reduction Formula

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Piecewise Definite integration

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 8.11

Line : 49

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