NDA Maths · Indefinite Integration

Foundations & Standard Forms

Integration is differentiation run backwards: given a rate of change, recover the function — plus an unknown constant C that no derivative can pin down.

All Indefinite Integration notes NDA Maths strategy

Why this matters

Before any technique, three reflexes carry the whole chapter: an indefinite integral is a FAMILY of functions (the +C), the standard-formula table must be instant recall, and most NDA integrands are simplified with exponent/log laws BEFORE a formula applies. 13 PYQs sit directly here — exponential bases, the eˡⁿ-collapse trick, completing the square, the eˣ[f+f′] pattern, and the paired eˣ·trig integrals — and these reflexes underpin all 40 questions in the chapter.

Concept 1 of 9

Antiderivative and the Constant of Integration

Intuition

Differentiation turns a function into its slope. Integration runs that backwards — given the slope everywhere, find the function. But a constant has zero slope, so ANY vertical shift of a correct answer is also correct. That unknown shift is the +C.

Definition

A function $F$ is an antiderivative of $f$ if $F'(x) = f(x)$ . The indefinite integral $\int f(x)\,dx = F(x) + C$ denotes the whole family of antiderivatives, where $C$ is an arbitrary constant. Because $\dfrac{d}{dx}[F(x) + C] = f(x)$ for every constant $C$ , the constant can never be recovered from $f$ alone — an extra condition (a boundary value) is needed to fix it.

Indefinite integral

\int f(x)\,dx = F(x) + C \quad\text{where}\quad F'(x) = f(x)

F(x)any one antiderivative of $f$
Carbitrary constant of integration

Visualization · the +C family of antiderivatives

Every curve is an antiderivative of f(x) = x. They differ only by the constant C — a vertical shift. At x = 1 the red tangents are all parallel (slope = f(1) = 1): same derivative, infinitely many curves. That is why every indefinite integral carries a + C.

Worked example

Verify that both

\sin x + 7

and

\sin x - 2

are antiderivatives of

\cos x

, and write the indefinite integral.

Differentiate the first: $\dfrac{d}{dx}(\sin x + 7) = \cos x + 0 = \cos x$ . ✓
Differentiate the second: $\dfrac{d}{dx}(\sin x - 2) = \cos x$ . ✓
They differ only by a constant, so the whole family is $\sin x + C$ .

Answer:

\displaystyle\int \cos x\,dx = \sin x + C

Never drop the +C on an indefinite integral

An indefinite integral with no

+C

is incomplete. NDA options are written so the 'no constant' version and a wrong-constant version both appear — only the form carrying

+C

(or

+k

) is correct.

Concept 2 of 9

The Standard-Formula Table

Intuition

About a dozen integrals are the alphabet of the whole chapter. Every technique — substitution, parts, partial fractions — exists only to REDUCE a hard integral to one of these. Know them as reflexes, not look-ups.

Definition

The integrals you must recall instantly:

$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \neq -1$
$\int \dfrac{1}{x}\,dx = \ln|x| + C$
$\int e^x\,dx = e^x + C$ , and $\int a^x\,dx = \dfrac{a^x}{\ln a} + C$
$\int \sin x\,dx = -\cos x + C$ , $\int \cos x\,dx = \sin x + C$
$\int \sec^2 x\,dx = \tan x + C$ , $\int \csc^2 x\,dx = -\cot x + C$
$\int \sec x\tan x\,dx = \sec x + C$ , $\int \tan x\,dx = \ln|\sec x| + C$
$\int \dfrac{dx}{1+x^2} = \tan^{-1}x + C$ , $\int \dfrac{dx}{\sqrt{1-x^2}} = \sin^{-1}x + C$

Power rule (the most-used row)

\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)

n \neq -1the exclusion that makes $\int x^{-1} = \ln|x|$ a separate row

Worked example

Evaluate

\displaystyle\int\left(\sqrt{x} + \sec^2 x\right)dx

Write $\sqrt{x} = x^{1/2}$ and apply the power rule: $\int x^{1/2}\,dx = \dfrac{x^{3/2}}{3/2} = \dfrac{2}{3}x^{3/2}$ .
From the table, $\int \sec^2 x\,dx = \tan x$ .
Add and attach one constant.

Answer:

\dfrac{2}{3}x^{3/2} + \tan x + C

The power rule excludes $n=-1$

\int x^{-1}\,dx

is NOT

\dfrac{x^0}{0}

— that is undefined. It is the special row

\int \dfrac{1}{x}\,dx = \ln|x| + C

Concept 3 of 9

Linearity — Integrate Term by Term

Intuition

Integration is linear: a constant multiplier slides out, and a sum integrates piece by piece. This lets you break any polynomial or sum into table look-ups.

Definition

For constants $a, b$ :

\int\big(a\,f(x) + b\,g(x)\big)\,dx = a\int f(x)\,dx + b\int g(x)\,dx.

Only ONE constant of integration is needed for the whole expression — collect the separate constants into a single

C

at the end. Linearity does NOT extend to products or quotients:

\int f g \neq \int f \cdot \int g

Linearity of the integral

\int\big(a\,f(x)+b\,g(x)\big)\,dx = a\!\int\! f(x)\,dx + b\!\int\! g(x)\,dx

Worked example

Evaluate

\displaystyle\int\dfrac{2x^2 - 3}{x}\,dx

Split the quotient first (NOT a product rule): $\dfrac{2x^2-3}{x} = 2x - \dfrac{3}{x}$ .
Integrate term by term: $\int 2x\,dx = x^2$ and $\int \dfrac{3}{x}\,dx = 3\ln|x|$ .
Combine with one constant.

Answer:

x^2 - 3\ln|x| + C

You cannot split a product or a quotient like a sum

\int \dfrac{1}{x(x^2+1)}\,dx \neq \int\dfrac{1}{x}\,dx \cdot \int\dfrac{1}{x^2+1}\,dx

. Linearity is for SUMS only — products need substitution or partial fractions.

Concept 4 of 9

Simplify the Integrand First

Intuition

Many NDA integrals look terrifying until you apply one exponent or log law — then they collapse to a table form. Before reaching for a method, ask: can algebra make this a standard integral?

Definition

The collapsing identities the NDA tests most:

$e^{\ln u} = u$ — the exponential and natural log undo each other, so $\int e^{\ln(\tan x)}\,dx = \int \tan x\,dx$ .
$\ln(u^k) = k\ln u$ , so $e^{k\ln x} = x^k$ — a stacked log/exponent becomes a power.
A quotient like $\dfrac{P(x)}{x}$ splits into powers (linearity), and the $\dfrac{1}{x}$ term is exactly what forces a $\ln$ (or must vanish for a rational answer).

The collapse identity

e^{\ln u} = u \qquad e^{k\ln x} = x^{k}

Worked example

Evaluate

\displaystyle\int e^{3\ln x}\,dx

Use $3\ln x = \ln(x^3)$ , so $e^{3\ln x} = e^{\ln(x^3)} = x^3$ .
The integral is now a power: $\int x^3\,dx = \dfrac{x^4}{4}$ .
Attach the constant.

Answer:

\dfrac{x^4}{4} + C

From the bank · past-year question

Example 4Indefinite IntegrationEASY

What is

\int e^{\ln(\tan x)}\,dx

equal to?

[Q84 · Sep · 2018]

Resolve the exponent/log BEFORE you integrate

Students who integrate

e^{\ln(\tan x)}

as if the exponent were a variable get nonsense.

e^{\ln(\tan x)}

is just

\tan x

— simplify, then integrate.

Drill 2 more on simplify the integrand first

Concept 5 of 9

Exponential Bases — a to the x

Intuition

Any constant raised to the x integrates with the same a-to-the-x on top and a ln of the base on the bottom. The whole skill is recognising the base, even when it is disguised as e to the (x ln a) or as a product like (4e) to the 2x.

Definition

The base- $a$ rule:

\int a^x\,dx = \dfrac{a^x}{\ln a} + C \quad (a>0,\ a\neq 1).

Disguises to unmask:

e^{x\ln a} = a^x

; and

(kc)^{mx} = \big((kc)^m\big)^x

, so

(4e)^{2x} = \big(16e^2\big)^x

integrates as a base-

16e^2

exponential.

Exponential base rule

\int a^x\,dx = \dfrac{a^x}{\ln a} + C

athe constant base, $a>0,\ a\neq 1$
\ln anatural log of the base — the divisor

Worked example

Evaluate

\displaystyle\int 5^{x}\,dx

and

\displaystyle\int e^{x\ln 7}\,dx

First is the rule directly: $\int 5^x\,dx = \dfrac{5^x}{\ln 5} + C$ .
For the second, $e^{x\ln 7} = (e^{\ln 7})^x = 7^x$ .
So $\int e^{x\ln 7}\,dx = \int 7^x\,dx = \dfrac{7^x}{\ln 7} + C$ .

Answer:

\dfrac{5^x}{\ln 5} + C

and

\dfrac{7^x}{\ln 7} + C

From the bank · past-year question

Example 5Indefinite IntegrationEASY

What is

\displaystyle\int e^{x\ln a}\,dx

equal to?

[Q94 · Apr · 2019]

Divide by $\ln a$ , not by $a$

\int a^x\,dx = \dfrac{a^x}{\ln a}

, never

\dfrac{a^x}{a}

and never

a^x\ln a

(that is the derivative). The

\ln a

lives in the denominator.

Drill 1 more on exponential bases — a to the x

Concept 6 of 9

Completing the Square for Quadratic Denominators

Intuition

When the denominator is an irreducible quadratic with no helpful numerator, force it into the (something)-squared-plus-constant shape. That instantly matches the arctan standard form.

Definition

Drive any $ax^2+bx+c$ into the form $(x-h)^2 + k^2$ by completing the square, then use:

\int \dfrac{dx}{x^2 + k^2} = \dfrac{1}{k}\tan^{-1}\!\Big(\dfrac{x}{k}\Big) + C.

If the leading coefficient is not 1, factor it out of the whole denominator first so the

x^2

coefficient is 1 before completing the square.

Arctan standard form

\int \dfrac{dx}{x^2+k^2} = \dfrac{1}{k}\tan^{-1}\!\Big(\dfrac{x}{k}\Big) + C

Worked example

Evaluate

\displaystyle\int \dfrac{dx}{x^2 + 4x + 13}

Complete the square: $x^2+4x+13 = (x+2)^2 + 9 = (x+2)^2 + 3^2$ .
Substitute $t = x+2$ (so $dt = dx$ ): the integral is $\int \dfrac{dt}{t^2 + 3^2}$ .
Apply the arctan form with $k=3$ : $\dfrac{1}{3}\tan^{-1}\!\big(\tfrac{t}{3}\big)$ , then put $t = x+2$ back.

Answer:

\dfrac{1}{3}\tan^{-1}\!\Big(\dfrac{x+2}{3}\Big) + C

From the bank · past-year question

Example 6Indefinite IntegrationMODERATE

What is

\displaystyle\int \dfrac{dx}{2x^{2} - 2x + 1}

equal to ?

[Q81 · Sep · 2019]

Factor out the leading coefficient first

For

\int\dfrac{dx}{2x^2-2x+1}

, pull the 2 out:

2\big(x^2 - x + \tfrac12\big)

, THEN complete the square inside. Skipping this gives the wrong

k

and a wrong coefficient.

Concept 7 of 9

The e-to-the-x Times f-plus-f-prime Pattern

Intuition

Whenever an integrand is e to the x multiplied by a function PLUS its own derivative, the answer is simply e to the x times that function. It is the product rule run backwards, and the NDA hides it inside long bracketed expressions.

Definition

The pattern:

\int e^x\big(f(x) + f'(x)\big)\,dx = e^x f(x) + C.

It works because

\dfrac{d}{dx}\big(e^x f(x)\big) = e^x f(x) + e^x f'(x) = e^x\big(f + f'\big)

. The skill is spotting the split — group the bracket into a function and its derivative. A related trick: when two integrals are easier to ADD than to do alone (

I_1 + I_2

), combine the integrands first.

Reverse product rule

\int e^x\big(f(x)+f'(x)\big)\,dx = e^x f(x) + C

Worked example

Evaluate

\displaystyle\int e^x\big(\tan x + \sec^2 x\big)\,dx

Spot the split: let $f(x) = \tan x$ . Then $f'(x) = \sec^2 x$ .
The integrand is exactly $e^x\big(f + f'\big)$ .
Apply the pattern: the answer is $e^x f(x)$ .

Answer:

e^x \tan x + C

From the bank · past-year question

Example 7Indefinite IntegrationMODERATE

What is

\int e^x\{1+\ln x + x\ln x\}\,dx

equal to?

[Q75 · Sep · 2022]

The whole bracket must be $f + f'$

Identify

f

so that the LEFTOVER terms are exactly

f'

. If they are not, the shortcut does not apply and you fall back to substitution or parts. Check by differentiating your proposed

e^x f(x)

Drill 1 more on the e-to-the-x times f-plus-f-prime pattern

Concept 8 of 9

Cyclic and Paired Integrals of e-to-the-x Times Trig

Intuition

Integrals of e to the x times sine or cosine come back to themselves after two by-parts steps, so they are best memorised as a matched pair. NDA passages then ask you to ADD, SUBTRACT, or READ these results rather than re-derive them.

Definition

The two standard results (each provable by parts — see Integration by Parts):

\int e^x\cos x\,dx = \dfrac{e^x(\cos x + \sin x)}{2} + C,

\int e^x\sin x\,dx = \dfrac{e^x(\sin x - \cos x)}{2} + C.

Verify either by differentiating the right side — a tool you already have. With

u = \int e^x\cos x\,dx

and

v = \int e^x\sin x\,dx

, note

\dfrac{du}{dx} = e^x\cos x

and

\dfrac{dv}{dx} = e^x\sin x

by the fundamental theorem — so

u+v = e^x\sin x = \dfrac{dv}{dx}

. Passages on a given antiderivative form

\big(\text{e.g. } U(x)V(x) - 3\ln|U+V|\big)

are solved by integrating, then matching the result piece-by-piece to read off

U

and

V

The matched pair

\int e^x\cos x\,dx = \tfrac{e^x(\cos x+\sin x)}{2}+C,\quad \int e^x\sin x\,dx = \tfrac{e^x(\sin x-\cos x)}{2}+C

Worked example

Using the standard pair, find

\displaystyle\int e^x(\cos x - \sin x)\,dx

By linearity, this is $\int e^x\cos x\,dx - \int e^x\sin x\,dx$ .
Subtract the pair: $\dfrac{e^x(\cos x+\sin x)}{2} - \dfrac{e^x(\sin x-\cos x)}{2} = \dfrac{e^x(2\cos x)}{2}$ .
Simplify.

Answer:

e^x \cos x + C

From the bank · past-year question

Example 8Indefinite IntegrationMODERATE

For the following two (02) items: Let

u=\int e^x\cos x\,dx

and

v=\int e^x\sin x\,dx

What is

u+v

equal to?

[Q85 · Sep · 2025]

du/dx is the integrand, not the other integral

u = \int e^x\cos x\,dx

, then

\dfrac{du}{dx} = e^x\cos x

(you undo the integral), NOT

-v

or any other integral. Differentiation cancels the integral sign directly.

Drill 3 more on cyclic and paired integrals of e-to-the-x times trig

Concept 9 of 9

Properties of an Antiderivative

Intuition

Some NDA items are reasoning questions: they ask whether an antiderivative inherits a property of its integrand — most often periodicity. The key fact is that integrating ADDS a linear term, which can destroy periodicity even when the integrand is periodic.

Definition

A function $g$ is periodic with period $p$ if $g(x+p) = g(x)$ . Integrating a periodic function need NOT give a periodic result. Example: $\sin^2 x = \dfrac{1-\cos 2x}{2}$ is periodic, but $\int \sin^2 x\,dx = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$ carries a $\dfrac{x}{2}$ term that grows without bound. So 'the integrand is periodic' is true, while 'the antiderivative is periodic' is false — two statements that look linked but are not.

Worked example

\displaystyle\int(1+\cos 2x)\,dx

a periodic function of

x

Integrate: $\int 1\,dx + \int \cos 2x\,dx = x + \dfrac{\sin 2x}{2} + C$ .
The $\sin 2x / 2$ part is periodic, but the bare $x$ term is not — it increases forever.
A sum of a periodic and a non-periodic (unbounded) term is not periodic.

Answer:No — the linear

x

term destroys periodicity, even though the integrand

1+\cos 2x

is periodic.

From the bank · past-year question

Example 9Indefinite IntegrationMODERATE

Let

f(x)

be an indefinite integral of

\sin^2 x

. Consider the following statements: Statement 1: The function

f(x)

satisfies

f(x+\pi)=f(x)

for all real

x

. Statement 2:

\sin^2(x+\pi)=\sin^2 x

for all real

x

. Which one of the following is correct in respect of the above statements?

[Q85 · Apr · 2017]

Two true facts can still give a false link

\sin^2(x+\pi)=\sin^2 x

' is true and 'the integrand is periodic' is true — but neither makes the ANTIDERIVATIVE periodic. Judge the statement about

f(x)

, not the one about the integrand, on its own.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (8)

Antiderivative and the Constant of Integration
Indefinite integral
$\int f(x)\,dx = F(x) + C \quad\text{where}\quad F'(x) = f(x)$
The Standard-Formula Table
Power rule (the most-used row)
$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
Linearity — Integrate Term by Term
Linearity of the integral
$\int\big(a\,f(x)+b\,g(x)\big)\,dx = a\!\int\! f(x)\,dx + b\!\int\! g(x)\,dx$
Simplify the Integrand First
The collapse identity
$e^{\ln u} = u \qquad e^{k\ln x} = x^{k}$
Exponential Bases — a to the x
Exponential base rule
$\int a^x\,dx = \dfrac{a^x}{\ln a} + C$
Completing the Square for Quadratic Denominators
Arctan standard form
$\int \dfrac{dx}{x^2+k^2} = \dfrac{1}{k}\tan^{-1}\!\Big(\dfrac{x}{k}\Big) + C$
The e-to-the-x Times f-plus-f-prime Pattern
Reverse product rule
$\int e^x\big(f(x)+f'(x)\big)\,dx = e^x f(x) + C$
Cyclic and Paired Integrals of e-to-the-x Times Trig
The matched pair
$\int e^x\cos x\,dx = \tfrac{e^x(\cos x+\sin x)}{2}+C,\quad \int e^x\sin x\,dx = \tfrac{e^x(\sin x-\cos x)}{2}+C$

Watch out for (9)

Never drop the +C on an indefinite integral
→ Antiderivative and the Constant of Integration
The power rule excludes $n=-1$
→ The Standard-Formula Table
You cannot split a product or a quotient like a sum
→ Linearity — Integrate Term by Term
Resolve the exponent/log BEFORE you integrate
→ Simplify the Integrand First
Divide by $\ln a$ , not by $a$
→ Exponential Bases — a to the x
Factor out the leading coefficient first
→ Completing the Square for Quadratic Denominators
The whole bracket must be $f + f'$
→ The e-to-the-x Times f-plus-f-prime Pattern
du/dx is the integrand, not the other integral
→ Cyclic and Paired Integrals of e-to-the-x Times Trig
Two true facts can still give a false link
→ Properties of an Antiderivative

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Indefinite IntegrationEASY

What is the value of

k

such that integration of

\frac{3x^2+8-4k}{x}

with respect to

x

, may be a rational function?

[Q90 · Apr · 2020]

Example 2Indefinite IntegrationMODERATE

p(x)=(4e)^{2x}

, then what is

\int p(x)\,dx

equal to?

[Q78 · Apr · 2020]

Example 3Indefinite IntegrationMODERATE

I_1=\int\frac{e^x\,dx}{e^x+e^{-x}}

and

I_2=\int\frac{dx}{e^{2x}+1}

, then what is

I_1+I_2

equal to?

[Q63 · Apr · 2022]

Example 4Indefinite IntegrationHARD

Let

2\displaystyle\int\dfrac{x^2-1}{\sqrt{x^2+1}}\,dx = U(x)V(x)-3\ln\{U(x)+V(x)\}+c

What is

|U^2(x)-V^2(x)|

equal to?

[Q99 · Sep · 2024]

Example 5Indefinite IntegrationMODERATE

What is

\int e^{(2\ln x+\ln x^2)}\,dx

equal to?

[Q100 · Apr · 2021]

Drill every past-year question on this subtopic

13 questions from the bank — paginated, with cart and Word-export support.

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