Properties of the Definite Integral MCQ - Practice Questions & Answers

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Properties of the Definite Integral (Part 2) - King's Property is considered one the most difficult concept.
48 Questions around this concept.

Solve by difficulty

Statement - I: The value of the integral $\int_{\pi /6}^{\pi /3}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\frac{\pi }{6}$

Statement - II : $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$

Statement - I is false; Statement - II is true

Statement - I is true ; Statement - II is true; Statement - II is a correct explanation for Statement-I

Statement - I is true ; Statement - II is true; Statement - II is not a correct explanation for Statement-I

Statement - I is true; Statement - II is false

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Concepts Covered - 2

Properties of the Definite Integral (Part 1)

Definite integrals have properties that relate to the limits of integration.

Property 1

$\int_{a}^{a}f(x)\;dx=0$

If the upper and lower limits of integration are the same, the integral is just a line and contains no area, hence the value is 0

Alternatively

$\\\mathrm{If\;\;\frac{d}{dx}\;F(x)=f(x),\;\;then}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{a}^{a}f(x)\;dx=\left [ F(x) \right ]^a_a}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=F(a)-F(a)}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0}$

Property 2

The value of the definite integral of a function over any particular interval depends on the function and the interval but not on the variable of the integration.

$\int_{a}^{b} f(x) d x=\int_{a}^{b} f(t) d t=\int_{a}^{b} f(y) d y$

$\\\mathrm{For\;example,\;\;}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{0}^{2}x^2\;dx=\left [ \frac{x^3}{3} \right ]^2_0}=\frac{2^3}{3}-\frac{0^3}{3}=\frac{2^3}{3}\\\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{0}^{2}t^2\;dt=\left [ \frac{t^3}{3} \right ]^2_0}=\frac{2^3}{3}-\frac{0^3}{3}=\frac{2^3}{3}\\\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{0}^{2}y^2\;dy=\left [ \frac{y^3}{3} \right ]^2_0}=\frac{2^3}{3}-\frac{0^3}{3}=\frac{2^3}{3}$

Property 3

If the limits of definite integral are interchanged, thenn its value changes by minus sign only.

$\\\mathrm{If\;\;\frac{d}{dx}\;F(x)=f(x),\;\;then}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{a}^{b}f(x)\;dx=\left [ F(x) \right ]^b_a}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=F(b)-F(a)}\\\\\mathrm{and,\;\;\;\;\;\;\int_{b}^{a}f(x)\;dx=\left [ F(x) \right ]^a_b}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left (F(a)-F(b) \right )}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=-\left (F(b)-F(a) \right )}\\\text{Hence,}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\int_{a}^{b}f(x)\;dx=-\int_{b}^{a}f(x)\;dx}$

Properties of the Definite Integral (Part 2) - King's Property

Property 4 (King's Property)

This is one of the most important properties of definite integration.

$\\\mathbf{\int_{a}^{b}f(x)\;dx=\int_{a}^{b}f(a+b-x)\;dx}$

Proof:

$\\\mathrm{In\;\;R.H.S,\;}\text { Put } t=a+b-x\\ \Rightarrow \;\;\;\;\;\;\;\;\;\;d x=-d t\\\text {Also, when } x=a, \text { then } t=b, \text { and when } x=b,\;t=a\\\\\therefore \;\;\;\;\;\;\;\;\text{R.H.S}=\int_{b}^{a}f(t)\;(-dt)\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=-\int_{b}^{a}f(t)\;dt\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=-\left (-\int_{a}^{b}f(t)\;dt \right )\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\int_{a}^{b}f(t)\;dt \\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\int_{a}^{b}f(x)\;dx \\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\text{L.H.S}$