NDA Mathematics · Formula sheet

Indefinite Integration formulas

23 formulas and 26 common traps for NDA Mathematics Indefinite Integration, grouped by subtopic.

Foundations & Standard Forms

Antiderivative and the Constant of Integration

Indefinite integral

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

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

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)

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

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

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

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

Properties of an Antiderivative

Integration and differentiation are inverse

\int F'(x)\,dx = F(x) + C \qquad \dfrac{d}{dx}\!\int f(x)\,dx = f(x)

$F'(x)$ a derivative; integrating it recovers $F$ up to a constant

Trigonometric Simplification Toolkit

The collapses you reach for most

1+\cos x = 2\cos^2\tfrac{x}{2},\quad 1-\cos x = 2\sin^2\tfrac{x}{2},\quad \sqrt{1\pm\sin 2x}=|\sin x\pm\cos x|

$\tfrac{x}{2}$ half-angle — appears whenever you collapse $1\pm\cos x$
$|\cdots|$ the root of a perfect square is a MODULUS; fix the sign on the given interval

Common traps

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.

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

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.

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.

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.

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.

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)

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.

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.

$1\pm\cos x$ (half-angle) vs $1\pm\cos 2x$ (power-reduction)

Different collapses:

1-\cos x = 2\sin^2\tfrac{x}{2}

but

1-\cos 2x = 2\sin^2 x

. Read the angle inside the cosine before choosing the factor — the wrong one halves or doubles the argument.

The root of a perfect square is a MODULUS

\sqrt{(\sin x-\cos x)^2} = |\sin x-\cos x|

, not

\sin x-\cos x

. Resolve the sign on the given interval: on

(\tfrac\pi4,\tfrac\pi2)

it is

+(\sin x-\cos x)

; on

(0,\tfrac\pi4)

it is

-(\sin x-\cos x)

. Dropping the modulus is the most common error on

\sqrt{1\pm\sin 2x}

problems.

$\sec\pm\tan$ — mind which way the half-angle shifts

\sec x+\tan x=\tan\!\left(\tfrac\pi4+\tfrac x2\right)

but

\sec x-\tan x=\tan\!\left(\tfrac\pi4-\tfrac x2\right)

. Useful check:

(\sec x+\tan x)(\sec x-\tan x)=\sec^2x-\tan^2x=1

Integration by Substitution

Learn this subtopic in the notes

Why Substitution Works — the Reverse Chain Rule

Substitution rule

\int f\big(g(x)\big)\,g'(x)\,dx = \int f(u)\,du,\quad u=g(x)

Algebraic and Composite Substitutions

Power-times-derivative shape

\int \big(g(x)\big)^n\,g'(x)\,dx = \dfrac{\big(g(x)\big)^{n+1}}{n+1} + C

The f-prime-over-f to Log Pattern

Log pattern

\int \dfrac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C

Trigonometric Substitutions and Identity Reductions

The divide-by-cos-squared move

\int \dfrac{dx}{a^2\sin^2 x + b^2\cos^2 x} = \int \dfrac{\sec^2 x\,dx}{a^2\tan^2 x + b^2} = \dfrac{1}{ab}\tan^{-1}\!\Big(\dfrac{a\tan x}{b}\Big) + C

Rationalising a Surd Denominator

Conjugate clears the surd

\dfrac{1}{\sqrt{x+a}-\sqrt{x+b}} = \dfrac{\sqrt{x+a}+\sqrt{x+b}}{a-b}

Spotting a Hidden Derivative

The x-to-the-x derivative

\dfrac{d}{dx}\,x^x = x^x(1+\ln x)

Common traps

Every x must disappear before you integrate in u

After substituting, the integral must be purely in

u

and

du

. A leftover

x

means

u

was chosen badly or a constant factor was mishandled — fix it before integrating.

Carry the sign from du

With

u=\cos x

du=-\sin x\,dx

— the minus sign is part of the substitution. Dropping it flips the answer's sign, and the wrong-sign option is always offered.

Adjust by a constant, never by a variable

If the numerator is

k

times

f'(x)

for a constant

k

, pull

k

out. But if it differs by a FUNCTION of

x

, the pattern does not apply — do not force it.

Which of $a$ , $b$ goes inside the arctan?

\int \dfrac{dt}{a^2t^2+b^2}

the coefficient of

t^2

ends up multiplying

t

inside:

\tan^{-1}\big(\tfrac{at}{b}\big)

, not

\tan^{-1}\big(\tfrac{bt}{a}\big)

. Check with

a=1

\int\frac{dt}{t^2+b^2} = \frac1b\tan^{-1}\frac{t}{b}

—

t

is divided by the constant term's root. A published key for this exact PYQ had

a

and

b

swapped; the

\tfrac{1}{ab}

outside is the same either way, so only the inside tells the options apart.

A square root forces an absolute value

\sqrt{(\sin x - \cos x)^2} = |\sin x - \cos x|

, and the sign depends on the given range of

x

. On

0<x<\tfrac{\pi}{4}

\cos x > \sin x

, so it equals

\cos x - \sin x

— read the interval before dropping the modulus.

Do not lose the 1 over (a minus b) factor

After conjugating, the constant denominator

a-b

stays as a multiplier on the whole integral. Forgetting it scales every coefficient wrong — the classic error on

\sqrt{x+1}-\sqrt{x-1}

(where

a-b=2

x-to-the-x is neither a power nor an exponential

\dfrac{d}{dx}x^x \neq x\cdot x^{x-1}

(power rule) and

\neq x^x\ln x

(exponential rule). Both the base and the exponent vary, so its derivative is

x^x(1+\ln x)

— derived via logarithmic differentiation.

Integration by Parts

Learn this subtopic in the notes

The By-Parts Formula and LIATE

Integration by parts

\int u\,dv = uv - \int v\,du

$u$ factor you differentiate (pick by LIATE)
$dv$ remaining factor, which you integrate to $v$

Integrating a Lone Logarithm

Integral of the logarithm

\int \ln x\,dx = x\ln x - x + C

Products and Telescoping Cancellations

The product-rule trade

\int u\,dv = uv - \int v\,du

Common traps

Choosing u backwards makes it worse

\int v\,du

is harder than where you started, you picked

u

and

dv

the wrong way round. LIATE almost always points to the right

u

— trust it.

ln x has no naive antiderivative

\int \ln x\,dx

is NOT

\dfrac{1}{x}

(that is the derivative) and NOT

\dfrac{(\ln x)^2}{2}

. It is

x\ln x - x

— derive it by parts if you ever forget it.

Simplify disguised factors before applying parts

e^{\ln x}

is just

x

;

e^{\ln(\tan x)}

is just

\tan x

. Collapse these FIRST — applying by-parts to the disguised form wastes a step and invites errors.

Integration by Partial Fractions

Learn this subtopic in the notes

Decomposition and the Cover-Up Method

Linear-factor decomposition

\dfrac{p(x)}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}

The Recurring 1 over x times x-to-the-n-plus-1 Family

Closed form for the family

\int \dfrac{dx}{x(x^n+1)} = \dfrac{1}{n}\ln\left|\dfrac{x^n}{x^n+1}\right| + C

Substitute First, Then Decompose

Trig-to-rational substitution

\int \dfrac{\sin\theta\,d\theta}{f(\cos\theta)} = -\int \dfrac{du}{f(u)},\quad u=\cos\theta

Express the Numerator via the Denominator and Its Derivative

Numerator as denom + derivative

N(x) = A\,D(x) + B\,D'(x)\ \Rightarrow\ \int \dfrac{N}{D}\,dx = A x + B\ln|D| + C

Common traps

Decompose only a PROPER fraction

If the numerator's degree is

\geq

the denominator's, do polynomial division FIRST, then decompose the remainder. Skipping this gives a wrong split.

The 1 over n out front is easy to lose

The coefficient is

\dfrac{1}{n}

, coming from

x^{n-1}dx = \tfrac1n d(x^n)

. For

n=7

the answer carries

\tfrac17

; the no-coefficient option is the planted distractor.

Carry the minus from du, and the chain factor

u=\cos\theta

gives

du=-\sin\theta\,d\theta

— the leading minus stays. And a factor like

3+4\cos\theta

contributes a

\tfrac14

to its log coefficient because

\dfrac{d}{du}(3+4u)=4

Watch the sign in the denominator's derivative

\dfrac{d}{dx}(2\cos x + 5\sin x) = -2\sin x + 5\cos x

— the cosine term's derivative is

-\sin

. A sign slip in the

2\times2

system swaps

A

and

B

, the exact distractor the paired items test.

Indefinite Integration formulas

Foundations & Standard Forms

Antiderivative and the Constant of Integration

The Standard-Formula Table

Linearity — Integrate Term by Term

Simplify the Integrand First

Exponential Bases — a to the x

Completing the Square for Quadratic Denominators

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

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

Properties of an Antiderivative

Trigonometric Simplification Toolkit

Common traps

Integration by Substitution

Why Substitution Works — the Reverse Chain Rule

Algebraic and Composite Substitutions

The f-prime-over-f to Log Pattern

Trigonometric Substitutions and Identity Reductions

Rationalising a Surd Denominator

Spotting a Hidden Derivative

Common traps

Integration by Parts

The By-Parts Formula and LIATE

Integrating a Lone Logarithm

Products and Telescoping Cancellations

Common traps

Integration by Partial Fractions

Decomposition and the Cover-Up Method

The Recurring 1 over x times x-to-the-n-plus-1 Family

Substitute First, Then Decompose

Express the Numerator via the Denominator and Its Derivative

Common traps

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