Core Syllabus Module

Definite Integration

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JEE Main 2024 (09 April Shift 1)

LEVELJEE Advanced

Let $lim_{n \to \infty} (\frac{n}{n ^{4} + 1} - \frac{2 n}{( n ^{2} + 1 ) n ^{4} + 1} + \frac{n}{n ^{4} + 16} - \frac{8 n}{( n ^{2} + 4 ) n ^{4} + 16} + \dots + \frac{n}{n ^{4} + n ^{4}} - \frac{2 n \cdot n ^{2}}{( n ^{2} + n ^{2} ) n ^{4} + n ^{4}})$ be $\frac{π}{k}$ , using only the principal values of the inverse trigonometric functions. Then $k^{2}$ is equal to ________

JEE Advanced 2018

LEVELJEE Advanced

Consider two straight lines, each of which is tangent to both the circle $x^{2} + y^{2} = \frac{1}{2}$ and the parabola $y^{2} = 4 x$ . Let these lines intersect at the point $Q$ . Consider the ellipse whose center is at the origin $O (0, 0)$ and whose semi-major axis is $O Q$ . If the length of the minor axis of this ellipse is $2$ , then which of the following statement(s) is (are) TRUE ?

* Multiple Correct Options

(A)

For the ellipse, the eccentricity is

\frac{1}{2}

and the length of the latus rectum is

1

(B)

For the ellipse, the eccentricity is

\frac{1}{2}

and the length of the latus rectum is

\frac{1}{2}

(C)

The area of the region bounded by the ellipse between the lines

x = \frac{1}{2}

and

x = 1

\frac{1}{4 2} (π - 2)

(D)

The area of the region bounded by the ellipse between the lines

x = \frac{1}{2}

and

x = 1

\frac{1}{16} (π - 2)

JEE Advanced 1998

LEVELJEE Advanced

Let $C_{1}$ and $C_{2}$ be the graphs of the functions $y = x^{2}$ and $y = 2 x, 0 \leq x \leq 1$ respectively. Let $C_{3}$ be the graph of a function $y = f (x), 0 \leq x \leq 1, f (0) = 0$ . For a point $P$ on $C_{1}$ , let the lines through $P$ , parallel to the axes, meet $C_{2}$ and $C_{3}$ at $Q$ and $R$ respectively (see figure.) If for every position of $P$ (on $C_{1}$ ), the areas of the shaded regions $O P Q$ and $O R P$ are equal, determine the function $f (x)$ .

JEE(ADVANCED)-202

LEVELJEE Advanced

Let the function $f : [1, \infty) \to R$ be defined by $f (t) = {\frac{( - 1 ) ^{n + 1} 2}{( 2 n + 1 - t )} - f (2 n - 1) + \frac{( t - ( 2 n - 1 ))}{2} f (2 n + 1); if t = 2 n - 1, n \in N; if 2 n - 1 < t < 2 n + 1, n \in N$ Define $g (x) = \int_{1}^{x} f (t) d t, x \in (1, \infty)$ . Let $α$ denote the number of solutions of the equation $g (x) = 0$ in the interval $(1, 8]$ and $β = lim_{x \to 1^{+}} \frac{g ( x )}{x - 1}$ . Then the value of $α + β$ is equal to ________.

JEE Advanced 1999

LEVELJEE Advanced

Let $f (x)$ be a continuous function given by $f (x) = {2 x, x^{2} + a x + b, ∣ x ∣ \leq 1 ∣ x ∣ > 1$ . Find the area of the region in the third quadrant bounded by the curves $x = - 2 y^{2}$ and $y = f (x)$ lying on the left of the line $8 x + 1 = 0$ .

JEE Advanced 2023

LEVELJEE Advanced

Let $f : [0, 1] \to [0, 1]$ be the function defined by $f (x) = \frac{x ^{3}}{3} - x^{2} + \frac{5}{9} x + \frac{17}{36}$ . Consider the square region $S = [0, 1] \times [0, 1]$ . Let $G = {(x, y) \in S : y > f (x)}$ be called the green region and $R = {(x, y) \in S : y < f (x)}$ be called the red region. Let $L_{h} = {(x, h) \in S : x \in [0, 1]}$ be the horizontal line drawn at a height $h \in [0, 1]$ . Then which of the following statements is(are) true? (A) There exists an $h \in [\frac{1}{4}, \frac{2}{3}]$ such that the area of the green region above the line $L_{h}$ equals the area of the green region below the line $L_{h}$ (B) There exists an $h \in [\frac{1}{4}, \frac{2}{3}]$ such that the area of the red region above the line $L_{h}$ equals the area of the red region below the line $L_{h}$ (C) There exists an $h \in [\frac{1}{4}, \frac{2}{3}]$ such that the area of the green region above the line $L_{h}$ equals the area of the red region below the line $L_{h}$ (D) There exists an $h \in [\frac{1}{4}, \frac{2}{3}]$ such that the area of the red region above the line $L_{h}$ equals the area of the green region below the line $L_{h}$

* Multiple Correct Options

The greatest integer less than or equal to $\int_{1}^{2} lo g_{2} (x^{3} + 1) d x + \int_{1}^{l o g_{2} 9} (2^{x} - 1)^{1/3} d x$ is ______.

JEE Advanced 2005

LEVELJEE Advanced

If $4 a^{2} 4 b^{2} 4 c^{2} 4 a 4 b 4 c 111 f (- 1) f (1) f (2) = 3 a^{2} + 3 a 3 b^{2} + 3 b 3 c^{2} + 3 c, f (x)$ is a quadratic function and its maximum value occurs at a point $V$ . $A$ is a point of intersection of $y = f (x)$ with $x$ -axis and point $B$ is such that chord $A B$ subtends a right angle at $V$ . Find the area enclosed by $f (x)$ and chord $A B$ .

JEE Advanced 2014

LEVELJEE Advanced

Following question has matching lists. Match list I with List II.

List-I

(P)

The number of polynomials

f (x)

with non-negative integer coefficients of degree

\leq 2

, satisfying

f (0) = 0

and

\int_{0}^{1} f (x) d x = 1

, is

(Q)

The number of points in the interval

[- 13, 13]

at which

f (x) = sin (x^{2}) + cos (x^{2})

attains its maximum value, is

(R)

\int_{- 2}^{2} \frac{3 x ^{2}}{1 + e ^{x}} d x

equals

(S)

\frac{\int _{- 1/2}^{1/2} c o s 2 x l o g ( \frac{1 + x}{1 - x} ) d x}{\int _{0}^{1/2} c o s 2 x l o g ( \frac{1 + x}{1 - x} ) d x}

List-II

Let the functions $f : R \to R$ and $g : R \to R$ be defined by $f (x) = e^{x - 1} - e^{- ∣ x - 1∣}$ and $g (x) = \frac{1}{2} (e^{x - 1} + e^{1 - x})$ . Then the area of the region in the first quadrant bounded by the curves $y = f (x)$ , $y = g (x)$ and $x = 0$ is

(A)

(2 - 3) + \frac{1}{2} (e - e^{- 1})

(B)

(2 + 3) + \frac{1}{2} (e - e^{- 1})

(C)

(2 - 3) + \frac{1}{2} (e + e^{- 1})

(D)

(2 + 3) + \frac{1}{2} (e + e^{- 1})

JEE Advanced 2019

LEVELJEE Advanced

Let $f : R \to R$ be given by $f (x) = (x - 1) (x - 2) (x - 5)$ . Define $F (x) = \int_{0}^{x} f (t) d t, x > 0$ . Then which of the following options is/are correct ?

* Multiple Correct Options

(A)

F has a local minimum at

x = 1

(B)

F has a local maximum at

x = 2

(C)

F (x) \neq = 0

for all

x \in (0, 5)

(D)

F has two local maxima and one local minimum in

(0, \infty)

JEE Advanced 2010

LEVELJEE Advanced

Match the statement in Column-I with the values in Column-II

List-I

(P)

A line from the origin meets the lines

\frac{x - 2}{1} = \frac{y - 1}{- 2} = \frac{z + 1}{1}

and

\frac{x - \frac{8}{3}}{2} = \frac{y + 3}{- 1} = \frac{z - 1}{1}

P

and

Q

respectively. If length

P Q = d

, then

d^{2}

(Q)

The values of

x

satisfying

tan^{- 1} (x + 3) - tan^{- 1} (x - 3) = sin^{- 1} (\frac{3}{5})

are

(R)

Non-zero vectors

a, b

and

c

satisfy

a \cdot b = 0

(b - a) \cdot (b + c) = 0

and

2∣ b + c ∣ = ∣ b - a ∣

. If

a = μ b + 4 c

, then the possible values of

μ

are

(S)

Let

f

be the function on

[- π, π]

given by

f (0) = 9

and

f (x) = \frac{s i n ( \frac{9 x}{2} )}{s i n ( \frac{x}{2} )}

for

x \neq = 0

. The value of

\frac{2}{π} \int_{- π}^{π} f (x) d x

List-II

Comprehension Passage

Let

F : R \to R

be a thrice differentiable function. Suppose that

F (1) = 0, F (3) = - 4

and

F (x) < 0

for all

x \in (1, 3)

. Let

f (x) = x F (x)

for all

x \in R

Question 1:

The correct statement(s) is(are)

* Multiple Correct Options

(A)

f^{'} (1) < 0

(B)

f (2) < 0

(C)

f^{'} (x) \neq = 0

for any

x \in (1, 3)

(D)

f^{'} (x) = 0

for some

x \in (1, 3)

Question 2:

If $\int_{1}^{3} x^{2} F^{'} (x) d x = - 12$ and $\int_{1}^{3} x^{3} F^{''} (x) d x = 40$ , then the correct expression(s) is (are)

* Multiple Correct Options

(A)

9 f^{'} (3) + f^{'} (1) - 32 = 0

(B)

\int_{1}^{3} f (x) d x = 12

(C)

9 f^{'} (3) - f^{'} (1) + 32 = 0

(D)

\int_{1}^{3} f (x) d x = - 12

JEE Advanced 2014

LEVELJEE Advanced

Comprehension Passage

Given that for each

a \in (0, 1)

lim_{h \to 0^{+}} \int_{h}^{1 - h} t^{- a} (1 - t)^{a - 1} d t

exists. Let this limit be

g (a)

. In addition, it is given that the function

g (a)

is differentiable on

(0, 1)

Question 1:

The value of $g (\frac{1}{2})$ is

(A)

π

(B)

2 π

(C)

\frac{π}{2}

(D)

\frac{π}{4}

Question 2:

The value of $g^{'} (\frac{1}{2})$ is

(A)

\frac{π}{2}

(B)

π

(C)

- \frac{π}{2}

(D)

JEE Main 2026 (22 January Shift 2)

LEVELJEE Advanced

Let $[\cdot]$ be the greatest integer function. If $α = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) d x$ , then $\frac{1}{π} \int_{0}^{α π} (\frac{s i n ^{2} θ}{s i n ^{6} θ + c o s ^{6} θ}) d θ$ is equal to .........

JEE Advanced 2008

LEVELJEE Advanced

Comprehension Passage

Consider the functions defined implicitly by the equation

y^{3} - 3 y + x = 0

on various intervals in the real line. If

x \in (- \infty, - 2) \cup (2, \infty)

, the equation implicitly defines a unique real valued differentiable function

y = f (x)

. If

x \in (- 2, 2)

, the equation implicitly defines a unique real valued differentiable function

y = g (x)

satisfying

g (0) = 0

Question 1:

If $f (- 102) = 22$ , then $f^{''} (- 102) =$

(A)

\frac{4 2}{7 ^{3} 3 ^{2}}

(B)

- \frac{4 2}{7 ^{3} 3 ^{2}}

(C)

\frac{4 2}{7 ^{3} 3}

(D)

- \frac{4 2}{7 ^{3} 3}

Question 2:

The area of the region bounded by the curve $y = f (x)$ , the x-axis, and the lines $x = a$ and $x = b$ , where $- \infty < a < b < - 2$ , is

(A)

\int_{a}^{b} \frac{x}{3 ( f ( x ) ^{2} - 1 )} d x + b f (b) - a f (a)

(B)

- \int_{a}^{b} \frac{x}{3 ( f ( x ) ^{2} - 1 )} d x + b f (b) - a f (a)

(C)

\int_{a}^{b} \frac{x}{3 ( f ( x ) ^{2} - 1 )} d x - b f (b) + a f (a)

(D)

- \int_{a}^{b} \frac{x}{3 ( f ( x ) ^{2} - 1 )} d x - b f (b) + a f (a)

Question 3:

$\int_{- 1}^{1} g^{'} (x) d x =$

(A)

2 g (- 1)

(B)

(C)

- 2 g (1)

(D)

2 g (1)

JEE Main 2021 (20 July Shift 1)

LEVELJEE Advanced

Let $T$ be the tangent to the ellipse $E : x^{2} + 4 y^{2} = 5$ at the point $P (1, 1)$ . If the area of the region bounded by the tangent $T$ , ellipse $E$ , lines $x = 1$ and $x = 5$ is $α 5 + β + γ cos^{- 1} (\frac{1}{5})$ , then $∣ α + β + γ ∣$ is equal to

JEE Main 2021 (24 February Shift 1)

LEVELJEE Advanced

The area (in sq. units) of the part of the circle $x^{2} + y^{2} = 36$ , which is outside the parabola $y^{2} = 9 x$ , is :

(A)

24 π + 33

(B)

12 π + 33

(C)

12 π - 33

(D)

24 π - 33

JEE Main 2022 (26 June Shift 1)

LEVELJEE Advanced

The value of the integral $\frac{48}{π ^{4}} \int_{0}^{π} x^{2} (\frac{3 π x ^{2}}{2} - x^{3}) \frac{s i n x}{1 + c o s ^{2} x} d x$ is equal to ______.

JEE Main 2025 (January)

LEVELJEE Advanced

If the area of the larger portion bounded between the curves $x^{2} + y^{2} = 25$ and $y = ∣ x - 1∣$ is $\frac{1}{4} (bπ + c)$ b, $c \in N,$ then $b + c$ is equal to

Micro-Topic Strategy Matrix

Fundamental Theorem & Properties of Definite Integrals

Newton-Leibniz & Reduction Formulas

Definite Integral as a Limit of a Sum

Area Bounded by Curves

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The greatest integer less than or equal to ∫12​log2​(x3+1)dx+∫1log2​9​(2x−1)1/3dx is ______.

Following question has matching lists. Match list I with List II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be given by f(x)=(x−1)(x−2)(x−5). Define F(x)=∫0x​f(t)dt,x>0. Then which of the following options is/are correct ?

Match the statement in Column-I with the values in Column-II

List-I

List-II

Comprehension Passage

The correct statement(s) is(are)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∫13​x2F′(x)dx=−12 and ∫13​x3F′′(x)dx=40, then the correct expression(s) is (are)

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of g(21​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of g′(21​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let [⋅] be the greatest integer function. If α=∫064​(x1/3−[x1/3])dx, then π1​∫0απ​(sin6θ+cos6θsin2θ​)dθ is equal to .........

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(−102​)=22​, then f′′(−102​)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The area of the region bounded by the curve y=f(x), the x-axis, and the lines x=a and x=b, where −∞<a<b<−2, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∫−11​g′(x)dx=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The area (in sq. units) of the part of the circle x2+y2=36, which is outside the parabola y2=9x, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of the integral π448​∫0π​x2(23πx2​−x3)1+cos2xsinx​dx is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the area of the larger portion bounded between the curves x2+y2=25 and y=∣x−1∣ is 41​(bπ+c) b, c∈N, then b+c is equal to

The greatest integer less than or equal to $\int_{1}^{2} lo g_{2} (x^{3} + 1) d x + \int_{1}^{l o g_{2} 9} (2^{x} - 1)^{1/3} d x$ is ______.

Let $f : R \to R$ be given by $f (x) = (x - 1) (x - 2) (x - 5)$ . Define $F (x) = \int_{0}^{x} f (t) d t, x > 0$ . Then which of the following options is/are correct ?

If $\int_{1}^{3} x^{2} F^{'} (x) d x = - 12$ and $\int_{1}^{3} x^{3} F^{''} (x) d x = 40$ , then the correct expression(s) is (are)

The value of $g (\frac{1}{2})$ is

The value of $g^{'} (\frac{1}{2})$ is

Let $[\cdot]$ be the greatest integer function. If $α = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) d x$ , then $\frac{1}{π} \int_{0}^{α π} (\frac{s i n ^{2} θ}{s i n ^{6} θ + c o s ^{6} θ}) d θ$ is equal to .........

If $f (- 102) = 22$ , then $f^{''} (- 102) =$

The area of the region bounded by the curve $y = f (x)$ , the x-axis, and the lines $x = a$ and $x = b$ , where $- \infty < a < b < - 2$ , is

$\int_{- 1}^{1} g^{'} (x) d x =$

The area (in sq. units) of the part of the circle $x^{2} + y^{2} = 36$ , which is outside the parabola $y^{2} = 9 x$ , is :

The value of the integral $\frac{48}{π ^{4}} \int_{0}^{π} x^{2} (\frac{3 π x ^{2}}{2} - x^{3}) \frac{s i n x}{1 + c o s ^{2} x} d x$ is equal to ______.

If the area of the larger portion bounded between the curves $x^{2} + y^{2} = 25$ and $y = ∣ x - 1∣$ is $\frac{1}{4} (bπ + c)$ b, $c \in N,$ then $b + c$ is equal to