Indefinite Integration — JEE Mains Maths Formulas and Traps

Algebraic Substitution

Clearing a root

Mixed roots of x

x=t^{n},\quad dx=nt^{n-1}\,dt,\quad n=\operatorname{lcm}(\text{root orders})

Two linear factors: use their ratio

Ratio substitution

\int\frac{dx}{(x-a)^p(x+b)^{2-p}}=\frac{1}{(a+b)(1-p)}\left(\frac{x-a}{x+b}\right)^{1-p}+C

Taking out a power of x

Power rule for a bracket

\int f'(x)\,[f(x)]^{k}\,dx=\frac{[f(x)]^{k+1}}{k+1}+C,\quad k\neq-1

Common traps

Given values are in x

With

x=t^6

, the point

x=64

is

t=2

, not

t=64

. Convert the point to

t

, or the answer back to

x

, before fixing the constant.

Differentiate the ratio in full

For

t=\frac{2x+1}{2x+3}

,

dt=\frac{4\,dx}{(2x+3)^2}

, not

\frac{2\,dx}{(2x+3)^2}

. The constant on top is

2\cdot3-2\cdot1

, and a wrong constant scales the whole answer.

A root of a power

\sqrt[n]{x^mg}=x^{m/n}\sqrt[n]{g}

needs

x>0

when

n

is even; otherwise

\sqrt{x^2}=|x|

and a sign appears. The condition

x>0

in the question is what allows the step.

Rational Functions and Standard Forms

Learn this subtopic in the notes

Partial fractions

Cover-up rule

\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},\quad A=\frac{pa+q}{a-b}

Completing the square and x ± 1/x

Complete the square

\int\frac{dx}{(x+p)^2+a^2}=\frac1a\tan^{-1}\frac{x+p}{a}+C

Numerator through the denominator

Split the numerator

\int\frac{A\,g(x)+B\,g'(x)}{g(x)}\,dx=Ax+B\ln|g(x)|+C

Common traps

Divide before splitting

Partial fractions need the top's degree below the bottom's. For

\frac{x^2}{x^2-1}

, first write

1+\frac{1}{x^2-1}

; splitting straight away loses the 1.

Match the sign to the top

For

x^2+1

on top, put

t=x-\frac1x

; for

x^2-1

, put

t=x+\frac1x

. The other choice leaves no

dt

in the numerator.

Keep the leftover constant

A quadratic top needs three pieces:

A\,Q+B\,Q'+C

. With only the first two, the constant

C

is lost, and so is its

\int\frac{dx}{\sqrt Q}

log term.

Trigonometric Integrals

Learn this subtopic in the notes

Everything in tan x or cot x

Tan substitution

\int\frac{dx}{a^2\sin^2x+b^2\cos^2x}=\frac1{ab}\tan^{-1}\left(\frac{a\tan x}{b}\right)+C

Substituting sin x, cos x or sin x ± cos x

Sum substitution

(\sin x\pm\cos x)^2=1\pm\sin2x

Common traps

Cot brings a minus sign

With

t=\cot x

,

dt=-\csc^2x\,dx

. Every term of the answer changes sign; dropping the minus gives each coefficient the wrong sign.

Pick t by the top

For

\cos x-\sin x

on top, put

t=\sin x+\cos x

; for

\cos x+\sin x

, put

t=\sin x-\cos x

. The other choice leaves no

dt

.

Integration by Parts and Reverse Differentiation

Learn this subtopic in the notes

Integration by parts

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

The pattern eˣ(f + f′)

The eˣ pattern

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

Spotting a product or quotient derivative

Quotient in reverse

\int\frac{u'v-uv'}{v^2}\,dx=\frac{u}{v}+C

Common traps

Signs alternate

Repeated parts alternates the signs:

\int x^2\cos x\,dx=x^2\sin x+2x\cos x-2\sin x

. One wrong sign changes every value computed from the answer.

Which piece is f

In

e^x\left(\frac1x-\frac1{x^2}\right)

,

f=\frac1x

, because its derivative is the other piece. Taking

f=-\frac1{x^2}

needs

f'=\frac2{x^3}

, which is not there.

Differentiate the guess

A guess that is close is not an antiderivative. Differentiate it and compare term by term:

\ln x

and

x-1

agree at

x=1

and nowhere else, so one cannot stand in for the other.

Indefinite Integration formulas

Algebraic Substitution

Clearing a root

Two linear factors: use their ratio

Taking out a power of x

Common traps

Rational Functions and Standard Forms

Partial fractions

Completing the square and x ± 1/x

Numerator through the denominator

Common traps

Trigonometric Integrals

Everything in tan x or cot x

Substituting sin x, cos x or sin x ± cos x

Common traps

Integration by Parts and Reverse Differentiation

The pattern eˣ(f + f′)

Spotting a product or quotient derivative

Common traps

More JEE Mains Maths formula sheets