JEE Mains Maths · Indefinite Integration

Rational Functions and Standard Forms

Splitting a fraction into partial fractions, completing a square to reach a standard integral, and writing a numerator through the denominator and its derivative.

All Indefinite Integration notes JEE Mains Maths strategy

Why this matters

Eleven PYQs, six of them multiple choice, and one from 2026. Four split a fraction into partial fractions, three of them after putting t = x², t = tan x or t = x eˣ; three complete a square or use x ± 1/x to reach a standard form; four write the numerator as a multiple of the denominator plus a multiple of its derivative. Three ideas cover the page.

Concept 1 of 3: Partial fractions

A fraction with a factored denominator is a sum of simpler fractions, one for each factor, and each of those integrates to a log or an inverse tangent. For distinct linear factors, each coefficient is found by covering its factor and putting in the root. Many questions hide the fraction: substitute first, then split.

Definition

If the top's degree is not below the bottom's, divide first.
Distinct linear factors: $\frac{A}{x-a}$ for each, with $A$ the cover-up value at $x=a$ .
A factor that does not split, such as $x^2+1$ , takes $\frac{Bx+C}{x^2+1}$ .
A function of $x^2$ , $\tan x$ or $xe^x$ : put $t$ equal to it first, then split in $t$ .

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}

Worked example

Find

\int\frac{3x+5}{x^2-1}\,dx

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 2 Apr 2026 Shift 2 · Q69Moderate

Example 1 · Indefinite Integration · Rational Functions and Standard Forms

Let

f(x) = \int\left( \frac{16x+ 24}{x^{2}+ 2x- 15} \right)dx

. If

f(4) = 14\log_{e}(3)

and

f(7) =\log_{e}\left( 2^{\alpha}\cdot3^{\beta} \right),\alpha,\beta\in N

, then

\alpha+\beta

is equal to:

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.

Drill 3 more on partial fractions

Concept 2 of 3: Completing the square and x ± 1/x

A quadratic that does not factor is a square plus a constant, and one over that is an inverse tangent. Complete the square first and the standard table does the rest. When the top is

x^2\pm1

and the bottom is a quartic with no odd powers, divide by

x^2

: the top becomes the derivative of

x\mp\frac1x

Definition

$\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa$ ; $\int\frac{dx}{x^2-a^2}=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|$ .
$\int\frac{dx}{\sqrt{x^2+a^2}}=\ln\left|x+\sqrt{x^2+a^2}\right|$ ; $\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac xa$ .
$\int\sqrt{x^2+a^2}\,dx=\frac x2\sqrt{x^2+a^2}+\frac{a^2}2\ln\left|x+\sqrt{x^2+a^2}\right|$ .
For $\frac{x^2\pm1}{x^4+kx^2+1}$ , divide by $x^2$ and put $t=x\mp\frac1x$ . Higher powers work the same way: $t=x^3+\frac1{x^3}$ .

Complete the square

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

Worked example

Find

\int\frac{dx}{x^2+4x+13}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 27 Jan 2024 · Q164Moderate

Example 2 · Indefinite Integration · Rational Functions and Standard Forms

The integral

\int\frac{\left( x^{8}-x^{2} \right)dx}{\left( x^{12}+ 3x^{6}+ 1 \right)\tan^{- 1}\left( x^{3}+\frac{1}{x^{3}} \right)}

is equal to :

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.

Drill 2 more on completing the square and x ± 1/x

Concept 3 of 3: Numerator through the denominator

Write the numerator as a multiple of the denominator plus a multiple of its derivative. The first part integrates to a multiple of

x

, and the second to a log. For a quadratic under a root, the same split turns the integral into a root, a log and

\int\sqrt Q

, each from the table.

Definition

$\frac{a\sin x+b\cos x}{c\sin x+d\cos x}$ or $\frac{ae^x+be^{-x}}{ce^x+de^{-x}}$ : write the top as $A\,g+B\,g'$ , where $g$ is the bottom.
Match the coefficients of $\sin x$ and $\cos x$ (or $e^x$ and $e^{-x}$ ) to find $A$ and $B$ .
$\frac{px+q}{Q}$ or $\frac{px+q}{\sqrt Q}$ : top $=A\,Q'+B$ .
$\frac{\text{quadratic}}{\sqrt Q}$ : top $=A\,Q+B\,Q'+C$ .

Split the numerator

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

Worked example

Find

\int\frac{\sin x}{\sin x+\cos x}\,dx

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 24 Jan 2025 · Q150Moderate

Example 3 · Indefinite Integration · Rational Functions and Standard Forms

\int\frac{2x^{2}+ 5x+ 9}{\sqrt{x^{2}+x+ 1}}dx=x\sqrt{x^{2}+x+ 1}+\alpha\sqrt{x^{2}+x+ 1}+

\beta\log_{e}\left| x +\frac{1}{2}+\sqrt{x^{2}+ x + 1} \right|+C

, where C is the constant of integration, then

\alpha+ 2\beta

is equal to

\_\_\_\_

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.

Drill 3 more on numerator through the denominator

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 (3)

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

Watch out for (3)

Divide before splitting
→ Partial fractions
Match the sign to the top
→ Completing the square and x ± 1/x
Keep the leftover constant
→ Numerator through the denominator

Test yourself on Indefinite Integration

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