Piecewise Definite integration MCQ - Practice Questions & Answers

How to Get Admission in NITs through JEE Main 2025 - Choose Best NIT in India for BTech

Quick Facts

Piecewise Definite integration is considered one the most difficult concept.
30 Questions around this concept.

Solve by difficulty

EASY

The integral

$\int_{0 }^{\pi }\sqrt{1+4\sin ^{2}\frac{x}{2}-4\sin \frac{x}{2}}dx$

equals:

$4\sqrt{3}-4$

$4\sqrt{3}-4-\frac{\pi }{3}$

$\pi -4$

$\frac{2\pi }{3}-4-4\sqrt{3}$

View Solution

Concepts Covered - 1

Piecewise Definite integration

Property 5

$\mathbf{\int_{a}^{b} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{b} f(x) d x\;\;}\text{where }c\in\mathbb{R}$

This property is useful when function is in the form of piecewise or discontinuous or non-differentiable at x = c in (a, b).

$\\\mathrm{Let\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{d}{dx}(F(x))=f(x)}\\\\\mathrm{\therefore \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{a}^{c}f(x)\;dx+\int_{c}^{b}f(x)\;dx}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left.F(x)\right|_{a} ^{c}+\left.F(x)\right|_{c} ^{b}}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=F(c)-F(a)+F(b)-F(c)}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=F(b)-F(a)}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\int_{a}^{b}f(x)\;dx}$

The above property can be also generalized into the following form

$\\\mathbf{\int_{a}^{b} f(x) d x=\int_{a}^{c_{1}} f(x) d x+\int_{c_{1}}^{c_{2}} f(x) d x+\ldots+\int_{c_{n}}^{b} f(x) d x}\\\text { where, } \quad a<c_{1}<c_{2}<\ldots<c_{n-1}<c_{n}<b.$

Property 6

$\mathbf{\int_{0}^{a} f(x)\; d x=\int_{0}^{a / 2} f(x)\; d x+\int_{0}^{a / 2} f(a-x) \;d x}$

Proof:

$\\\text{From the previous property,}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;}\int_{0}^{a} f(x) d x=\int_{0}^{a / 2} f(x) d x+\int_{a / 2}^{a} f(x) d x\\ {\text { Put } x=a-t \Rightarrow d x=-d t \text { in the second integral, }} \\ {\text { when } x=a / 2, \text { then } t=a / 2 \text { and when } x=a, \text { then } t=0}\\\\\therefore \;\;\;\;\;\;\quad \int_{0}^{a} f(x) d x=\int_{0}^{a / 2} f(x) d x+\int_{a / 2}^{0} f(a-t)(-d t)\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\int_{0}^{a / 2} f(x) d x+\int_{0}^{a / 2} f(a-t) d t}\\\\ \mathrm{\;\;}\;\;\;\;\;\;\quad\int_{0}^{a} f(x) d x=\int_{0}^{a / 2} f(x) d x+\int_{0}^{a / 2} f(a-x) d x$

Study it with Videos

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

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Books

Reference Books

Piecewise Definite integration

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 8.11

Line : 49

Clear your Basics with NCERT

NCERT Solutions for Class 11 Maths (2024-25)

July 04, 2019

NCERT Solutions for Class 12 Maths - Get Free PDFs 2024-25

June 10, 2019

E-books & Sample Papers

JEE Main Previous 10 Year Questions with Detailed Solutions (2015-2024)

78642+ Downloads

JEE Main 2025 Preparation Tips - Complete Strategy & Study Plan

1848+ Downloads

Get Answer to all your questions

JEE Main Articles

PDEU BTech Application Form 2024 (Out) - Apply Online here

Sep 19, 2024

Minimum Marks in JEE Main to qualify for JEE Advanced

Sep 19, 2024

JEE Main 2025 Syllabus PDF - Subject-Wise Detailed Syllabus

Sep 18, 2024

JEE Main Syllabus 2025 PDF (Physics, Chemistry, Maths)

Sep 19, 2024

JEE Main Exam Date 2025 - NTA JEE Mains Session 1 & 2 Exam Schedule

Sep 19, 2024

B.Tech Mining Engineering Salary Package 2024 - Highest, Average and Lowest Salary

Sep 19, 2024

BTech Mechanical Engineering Salary Packages 2024 - Highest, Average and Lowest Salary

Sep 19, 2024

BTech Chemical Engineering Salary Packages 2024 - Highest, Average and Lowest Salary

Sep 19, 2024

B.Tech/B.Arch Admissions OPEN

Amity University, Noida B.Tech Admissions 2024

Apply

Asia's Only University with the Highest US & UK Accreditation

JSS University Noida Admissions 2024

Apply

170+ Recruiters Including Samsung, Zomato, LG, Adobe and many more | Highest CTC 47 LPA

Atlas SkillTech University | B.Tech Admissions

Apply

Hands on Mentoring and Code Coaching | Cutting Edge Curriculum with Real World Application

ITM SLS Baroda University B.Tech Admissions 2024

Apply

Approved by UGC, PCI, & COA. Best Innovative Private Institute in Gujarat. 600+ Recruiters, 3000+ Students Placed, 1500+ Industrial Internship Opportunities, 250+ International Internships.