Analysis & Linear Algebra

Integration

12 previous-year questions →every option and the correct answer, free Practice this topic →one question at a time, no clock, check as you go

See where the marks are across every paper →

1. Riemann integration and criteria

Read this as a chapter →

Lec-39 Integrability

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

The Riemann criterion; watch for the role of upper and lower sums.

Lec-39 Riemann/Riemann Stieltjes Integral

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Riemann integrability — the criterion and the zoo free

In plain words

Chop the domain into thin strips and add up rectangles. It works exactly when the points where the function jumps are negligible in total length, which allows a great many of them.

Three equivalent criteria (f bounded on [a,b])

Riemann's: for every $\varepsilon$ there is a partition with $U(f,P) - L(f,P) < \varepsilon$ .
Darboux: upper integral = lower integral.
Lebesgue's: the set of discontinuities of f has measure zero.

Criterion 3 is the one to use in the exam. Measure-zero sets include every countable set and the Cantor set.

The zoo

f on [0,1]	Discontinuities	Riemann integrable?
Dirichlet $1_\mathbb{Q}$	everywhere	✗ (bounded but discontinuous on a set of measure 1)
Thomae (1/q at p/q, 0 at irrationals)	$\mathbb{Q}$ , countable	✓, integral 0
monotone f	countable	✓
f with finitely many discontinuities	finite	✓
$1_{Cantor set}$	Cantor set, measure 0	✓, integral 0
$1_{fat Cantor set}$ (positive measure)	the fat Cantor set	✗
sin(1/x), f(0)=0	{0}	✓ (bounded)
1/x, f(0)=0	unbounded	✗ as a proper integral

Operations

Sums, products, |f|, max/min of integrable functions are integrable.
Composition: g ∘ f with g continuous and f integrable ⇒ integrable. But f ∘ g with f integrable, g continuous can fail: Thomae ∘ (a suitable continuous g) gives the Dirichlet-like non-integrable function. Also $1_{(0,1]}$ ∘ Thomae $= 1_\mathbb{Q}$ : not integrable.
|f| integrable does not imply f integrable: f = 1 on $\mathbb{Q}, -1$ on irrationals. $- \int f = 0$ with f ≥ 0 continuous ⇒ f ≡ 0; with f merely integrable ⇒ f = 0 a.e.

Fundamental theorem

$- F(x) = \int_{a}$ ˣ f: F is continuous; F′(x) = f(x) at every continuity point of f.

If f = g′ with g differentiable and f integrable, then $\int_{a}$ ᵇ f = g(b) − g(a). (Derivatives need not be integrable: $g(x) = x^{2} \sin(1/x^{2})$ has unbounded g′.)

Improper integrals — the reference values

$\int_{1}^\infty$ dx/xᵖ converges iff $p > 1; \int_{0}^{1}$ dx/xᵖ converges iff $p < 1; \int_{0}^\infty \sin x/x$ dx $= \pi/2$ converges but not absolutely; $\int_{0}^\infty$ $e^{-x^{2}}$ $= \surd\pi/2; \Gamma(n+1) = n$ !.

Key takeaways

Bounded + discontinuities of measure zero ⇔ Riemann integrable.
Thomae vs Dirichlet is the canonical contrast.
Composition with a continuous outer function is safe; inner is not.

Riemann sums closing on the integralinteractivefree

Exam focus: Lebesgue's criterion (bounded + discontinuities of measure zero) decides Riemann integrability instantly. Know Dirichlet, Thomae, and that composition can destroy integrability.

Practice

Part C2025 Decembershow ▾

Let f:[0,1]→[0,1] be a monotonically increasing function, that is, a≤b implies f(a)≤f(b). For any

\alpha\in(0,1)

, let

L_\alpha

⁺=

lim_{x\to\alpha^{+}}f(x)

and

L_\alpha

⁻=

lim_{x\to\alpha^{-}}f(x)

denote the right hand and left hand limits respectively, provided they exist. For

\alpha\in(0,1)

, if

L_\alpha

⁺ and

L_\alpha

⁻ exist, define

U_\alpha=(L_\alpha

⁻,

L_\alpha

⁺) if

L_\alpha

⁻

<L_\alpha

⁺, and ∅ if

L_\alpha

⁺

\le{}L_\alpha

⁻. Which of the following statements are true?

A. $L_\alpha$ ⁺ and $L_\alpha$ ⁻ exist for every $\alpha\in(0,1)$ .✓ Correct
B.If f is surjective, then f is continuous.✓ Correct
C.If f is Riemann integrable, then f is continuous.
D.If the left and right hand limits exist at $\alpha,\beta\in(0,1), \alpha\ne\beta$ , then $U_\alpha\cap{}U_\beta=\emptyset$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Let f and g be real-valued Riemann integrable functions on [a, b] such that g([a, b]) ⊆ [a, b]. Which of the following statements are necessarily true?

A.The composition f ∘ g is Riemann integrable.
B.If g(x) ≠ 0 for each x ∈ [a, b], then f/g is Riemann integrable.
C.The positive square root $\sqrt{f^{2} + g^{2}}$ is Riemann integrable.✓ Correct
D.The composition f ∘ g is Riemann integrable, if both f and g are continuous.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

What is the value of the limit

lim_{n\to\infty}

(1/n)[(n + 1)(n + 2)⋯(n + n)]

^{1/n}

A.2/e
B.4/e✓ Correct
C. $log_{e}2 - 1$
D. $2 log_{e}2 - 1$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

f : [0, 1] \to \mathbb{R}

be a Riemann integrable function. Define

F(x) = \int_{0}

ˣ f(t)dt, ∀x ∈ [0, 1]. Which of the following statements are necessarily true?

A.F is Riemann integrable.✓ Correct
B.If F(x) = 0 for all x ∈ [0, 1], then f(x) = 0 for all x ∈ [0, 1].
C.F is uniformly continuous on [0, 1].✓ Correct
D.F is differentiable on (0, 1).

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

g : \mathbb{R} \to \mathbb{R}

be a continuous function. Define

f(x) = \int_{0}

ˣ(x − t)g(t)dt,

x \in \mathbb{R}

. Which of the following statements are true?

A.f(0) = 0✓ Correct
B.f′(0) exists and f′(0) = 0.✓ Correct
C.f″(0) exists and f″(0) = g(0).✓ Correct
D.f″(0) exists but f″(0) ≠ g(0).

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let

f : [0, 1] \to \mathbb{R}

be defined by

f(x) = \sin(x^{2})

. Let

A = \lim(n\to\infty) (\sum_(k=1)^n f(k/n) - n\int_{0}^{1}f(x)

dx). Which of the following statements is true?

A.A = 0
B.A = 1
C.A = sin(1)/2✓ Correct
D.A = sin(1/4)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

f : [0, 1] \to \mathbb{R}

be a monotonic function. Which of the following statements are true?

A.f is Riemann integrable on [0, 1].✓ Correct
B.The set of discontinuities of f cannot contain a non-empty open set.✓ Correct
C.f is Lebesgue measurable function.✓ Correct
D.f is Borel measurable function.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Suppose that

f : [-1, 1] \to \mathbb{R}

is continuous. Which of the following imply that f is identically zero on [−1, 1]?

A. $\int$ ₋ $_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0.✓ Correct
B. $\int$ ₋ $_{1}^{1} f(x)p(x)$ dx = 0 for all real polynomials p(x).✓ Correct
C. $\int$ ₋ $_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 odd.
D. $\int$ ₋ $_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 even.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

For a real number

\lambda

, consider the improper integrals

I_\lambda = \int_{0}^{1}

/(1 - x)^\lambda

and

K_\lambda = \int_{1}^\infty

/x^\lambda

. Which of the following statements are true?

A.There exists $\lambda$ such that $I_\lambda$ converges, but $K_\lambda$ does not converge.✓ Correct
B.There exists $\lambda$ such that $K_\lambda$ converges, but $I_\lambda$ does not converge.✓ Correct
C.There exists $\lambda$ such that $I_\lambda, K_\lambda$ both converge.
D.There exists $\lambda$ such that neither $I_\lambda$ nor $K_\lambda$ converges.✓ Correct

Solution

$I_\lambda$ converges iff $\lambda < 1; K_\lambda$ converges iff $\lambda > 1$ .

So they never converge together, and at $\lambda = 1$ neither does.

Save for spaced revision (free account)

Part Bshow ▾

The improper integral

\int_{0}^\infty

x^{p-1}

e^{-x}

dx converges if and only if

A.p > 0✓ Correct
B.p > 1
C.p ≥ 0
D.all real p

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Which of the following functions on [0, 1] are Riemann integrable?

A.The Dirichlet function $1_\mathbb{Q}$
B.Thomae's function✓ Correct
C.The indicator function of the Cantor set✓ Correct
D.f(x) = sin(1/x) for x > 0, f(0) = 0✓ Correct

Solution

(b) discontinuous only on $\mathbb{Q}; (c)$ discontinuous only on the Cantor set (measure zero); (d) bounded, one discontinuity.

(a) discontinuous everywhere.

Save for spaced revision (free account)

Part Cshow ▾

Which of the following functions on [0, 1] are Riemann integrable?

A.The Dirichlet function: f(x) = 1 for x rational, f(x) = 0 for x irrational.
B.Thomae's function: f(p/q) = 1/q in lowest terms, f(x) = 0 for x irrational, f(0) = 1.✓ Correct
C.f(x) = sin(1/x) for x ∈ (0, 1], f(0) = 0.✓ Correct
D.f(x) = 1/x for x ∈ (0, 1], f(0) = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

The limit

lim_{n\to\infty}

\sum_{k=1}^{n}

k/n^{2}

equals

A.0
B.1/2✓ Correct
C.1
D.2

Solution

$\sum{}k/n^{2} = (1/n)\sum(k/n)$ is a Riemann sum of $\int_{0}^{1} x$ dx = 1/2.

Directly: $n(n + 1)/(2n^{2}) \to 1/2$ .

Save for spaced revision (free account)

Part Bshow ▾

Let

f : [0,1] \to \mathbb{R}

be Riemann integrable and

g : \mathbb{R} \to \mathbb{R}

continuous. Which of the following is necessarily Riemann integrable on [0,1]?

A.f ∘ g (when defined)
B.g ∘ f✓ Correct
C.The function h with h(x) = 1 if $f(x) \in \mathbb{Q}$ and 0 otherwise
D.f restricted to $\mathbb{Q} \cap [0,1]$ , extended by 0

The worked solution and the per-option analysis are in Trap Analysis.

2. Improper integrals and convergence tests

Read this as a chapter →

Lec-45 Improper Integrals

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Both kinds of improperness — unbounded interval and unbounded integrand.

Lec-46 Convergence Test for Improper Integrals

NPTEL · A Basic Course in Real Analysis (IIT Kharagpur)

Comparison and the p-thresholds you need at sight.

Improper integrals — thresholds and tests free

In plain words

Something is infinite: either the function blows up, or the region runs on forever. Split so each piece has one problem, then compare with a power of $x$ . The two thresholds point opposite ways.

The two reference families

$\int_{1}^\infty$ dx/xᵖ converges ⇔ p > 1. $\int_{0}^{1}$ dx/xᵖ converges ⇔ p < 1.

So $\int_{0}^\infty$ dx/xᵖ never converges — the exam uses this constantly. Similarly $\int_{0}^{1}$ dx $/(1-x)^\lambda$ converges iff $\lambda < 1$ , while $\int_{1}^\infty$ dx $/x^\lambda$ converges iff $\lambda > 1$ : the two conditions are incompatible.

Logarithmic refinement: $\int_{2}^\infty$ dx/(x(log x)^p) converges ⇔ p > 1.

Tests

Comparison / limit comparison with xᵖ near each singular point.
Absolute convergence ⇒ convergence. The converse fails.
Dirichlet: $\int f g$ with f monotone → 0 and $\int{}g$ bounded converges — this is how $\int_{1}^\infty \sin x/x$ dx converges.
Split the integral at every singularity and at $\infty$ before testing; each piece must converge separately.

Standard values

Integral	Value
$\int_{0}^\infty$ $e^{-x^{2}}$ dx	$\surd\pi/2$
$\int_{0}^\infty \sin x / x$ dx	$\pi/2 ($ conditionally)
$\int_{0}^\infty$ $x^{s-1}e^{-x}$ dx	$\Gamma(s)$ , converges for s > 0
$\int_{0}^{1}$ $x^{a-1}(1-x)^{b-1}$ dx	$B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$ , needs a, b > 0
$\int_{0}^\infty$ dx $/(1+x^{2})$	$\pi/2$
$\int_{0}^\infty$ $x^{a-1}/(1+x)$ dx	$\pi/\sin(\pi{}a), 0 < a < 1$

The conditional-convergence gap

$\int_{1}^\infty \sin x/x$ dx converges but $\int_{1}^\infty$ |sin x|/x dx $= \infty$ . The integral analogue of $\sum(-1)^{n}/n$ .

Note also: convergence of $\int_{0}^\infty f$ does not force f(x) → 0. A function with tall thin spikes of area $2^{-n}$ at each integer converges yet is unbounded.

Key takeaways

p > 1 at infinity, p < 1 at the origin — and never both.
Conditional convergence is real for integrals too (sin x/x). $- \int{}f$ converging says nothing about f tending to 0.

The two exponents that decide most improper integralsinteractive

This visual is part of Trap Analysis

Why 1/x^p flips at p = 1 in opposite directions at 0 and at ∞ — and the integral that converges but not absolutely.

Unlock interactive visuals

Exam focus: Split at every singular point and at infinity, compare with 1/xᵖ, and distinguish convergence from absolute convergence — ∫ sin x/x is the standard example of the gap.

Practice

Part C2025 Decembershow ▾

Let F be the set of all functions

f : [0,\infty) \to [0,\infty)

that are Riemann integrable on [0,t] for all

t\in[0,\infty)

. Which of the following statements are necessarily true?

A.If f∈F is a continuous function and f(n)=0 for all positive integers n, then $\lim(x\to\infty)\int_{0}$ ˣf(t)dt exists in $\mathbb{R}$ .
B.If f∈F is uniformly continuous and $\lim(x\to\infty)\int_{0}$ ˣf(t)dt exists in $\mathbb{R}$ , then $\lim(x\to\infty)f(x)=0$ .✓ Correct
C.There exists a function f∈F such that $\lim(x\to\infty)\int_{0}$ ˣf(t)dt exists in $\mathbb{R}$ but $\lim(x\to\infty)f(x)\ne0$ .✓ Correct
D.If f∈F is Lebesgue measurable and $\lim(x\to\infty)\int_{0}$ ˣf(t)dt=0, then f=0 almost everywhere on $[0,\infty)$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Let

f : (0,\infty) \to (0,\infty)

be the function defined by

f(x) = \int_{0}

\sqrt{t}/(1+t^{2})

dt, where

\sqrt{t}

denotes the positive square root for t>0. Which of the following statements are true?

A.f is a uniformly continuous function.✓ Correct
B.f is a bounded function.✓ Correct
C.There exists $x\in(0,\infty)$ such that f(x)=0.
D.The derivative of f is continuous.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Consider the improper integrals

I = \int_(\pi/2)^\pi

/\sqrt{\sin x}

and, for

a \ge 0, I_{a} = \int_a^\infty

/(x\sqrt{1 + x^{2}})

. Which of the following statements are true?

A.The integral I is convergent✓ Correct
B.The integral I is not convergent
C.The integral $I_{a}$ converges for a = 1/2 but not for a = 0✓ Correct
D.The integral $I_{a}$ converges for all a ≥ 0

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

For which values of p does

\int_{0}^\infty

x^{p-1}/(1

+ x) dx converge?

A.0 < p < 1✓ Correct
B.p > 1
C.p > 0
D.no value of p

Solution

Near 0 the integrand ~ $x^{p-1}$ , needing p > 0; at $\infty$ it ~ $x^{p-2}$ , needing p < 1.

Save for spaced revision (free account)

Part Cshow ▾

Which of the following improper integrals converge?

A. $\int_{1}^\infty \sin(x)/x$ dx✓ Correct
B. $\int_{1}^\infty$ |sin(x)|/x dx
C. $\int_{0}^{1} 1/\sqrt{x}$ dx✓ Correct
D. $\int_{0}^{1} 1/x$ dx

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

The integral

\int_{1}^\infty (\sin x)/x

A.diverges
B.converges absolutely
C.converges conditionally✓ Correct
D.equals 0

The worked solution and the per-option analysis are in Trap Analysis.