JEE Mains Maths · Definite Integration

Odd, Even and Periodic Integrands

Integrals over an interval symmetric about 0, where odd parts vanish and even parts double; the 1/(1 + aᵍ⁽ˣ⁾) denominator that halves an even integrand; and integrals of periodic functions over many periods.

All Definite Integration notes JEE Mains Maths strategy

Why this matters

Twenty-five PYQs, most with limits −a to a. The integrand usually has an odd part that drops out, or a denominator like 1 + eˣ that pairs with its reflection to give 1. A few integrate a periodic function over many periods. Three ideas cover the page.

Concept 1 of 3: Odd parts vanish, even parts double

[-a,a]

an odd integrand contributes 0 and an even one contributes twice its integral on

[0,a]

. So split the integrand into its odd and even parts before doing anything else. A term like

\frac{x^3}{x^2+2|x|+1}

is odd, because everything but

x^3

depends only on

|x|

Definition

$f$ odd: $\int_{-a}^af=0$ . $f$ even: $\int_{-a}^af=2\int_0^af$ .
odd × even = odd; odd × odd = even.
$|x|$ , $x^2$ , $\cos x$ are even; $x$ , $\sin x$ , $\ln\big(x+\sqrt{x^2+1}\big)$ are odd.

Symmetric limits

\int_{-a}^{a}f=\begin{cases}0,&f\text{ odd}\\2\int_0^af,&f\text{ even}\end{cases}

Worked example

Evaluate

\int_{-1}^1(x^3+x^2+x)\,dx

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 2 · Q70Moderate

Example 1 · Definite Integration · Odd, Even and Periodic Integrands

The value of the integral

\int_{- 1}^{1} \left( \frac{x^{3}+ |x| + 1}{x^{2}+ 2|x| + 1} \right)dx

is equal to

Check the whole term

x|x|

is odd but

x^2|x|

is even. Decide the parity of each complete term, not of its pieces.

Drill 8 more on odd parts vanish, even parts double

Concept 2 of 3: A denominator 1 + bᵍ⁽ˣ⁾ with g odd

g

is odd and

h

is even, replacing

x

-x

turns

\frac{h(x)}{1+b^{g(x)}}

into

\frac{h(x)\,b^{g(x)}}{1+b^{g(x)}}

. The two add to

h(x)

, so the integral over

[-a,a]

is half of

\int_{-a}^ah

, which is

\int_0^ah

Definition

$\int_{-a}^a\frac{h(x)}{1+b^{g(x)}}dx=\int_0^ah(x)\,dx$ ( $h$ even, $g$ odd, $b>0$ ).
Common $g$ : $x$ , $\sin x$ , $x|x|$ , $x\cos x$ .
$\frac{e^x}{e^x+e^{-x}}=\frac1{1+e^{-2x}}$ is the same form.

Halving an even integrand

\int_{-a}^{a}\frac{h(x)}{1+b^{g(x)}}dx=\int_0^{a}h(x)\,dx

Worked example

Evaluate

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

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 25 · Q75Moderate

Example 2 · Definite Integration · Odd, Even and Periodic Integrands

The value of

\int_{-\pi/2}^{\pi/2} \frac{\cos^{2}x}{1 +3^{x}}dx

is:

The numerator must be even

An odd numerator over

1+b^{g}

does not halve. Split the numerator into even and odd parts and treat each by its own rule.

Drill 11 more on a denominator 1 + bᵍ⁽ˣ⁾ with g odd

Concept 3 of 3: Periodic integrands

f

repeats every

T

, every interval of length

T

gives the same integral, so

\int_0^{nT}f=n\int_0^Tf

. A relation like

f(x)+f(x+k)=c

makes

f

repeat every

2k

, and each stretch of length

2k

integrates to

ck

Definition

$f(x+T)=f(x)\Rightarrow\int_a^{a+nT}f=n\int_0^Tf$ .
Periods: $|\sin x|$ , $\sin^2x$ , $\sin^4x+\cos^4x$ : $\pi$ (the last is even $\frac\pi2$ ).
$f(x)+f(x+k)=c\Rightarrow$ period $2k$ and $\int_a^{a+2k}f=ck$ .

Many periods

\int_0^{nT}f(x)\,dx=n\int_0^{T}f(x)\,dx

Worked example

Evaluate

\int_0^{10\pi}|\sin x|\,dx

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 2 Apr 2026 Shift 2 · Q67Moderate

Example 3 · Definite Integration · Odd, Even and Periodic Integrands

The value of

\int_{0}^{20\pi} \left( \sin^{4}x +\cos^{4}x \right)dx

is equal to:

The right period

|\sin x|

repeats every

\pi

, not

2\pi

. Using the longer period halves the number of copies and halves the answer.

Drill 3 more on periodic integrands

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)

Odd parts vanish, even parts double
Symmetric limits
$\int_{-a}^{a}f=\begin{cases}0,&f\text{ odd}\\2\int_0^af,&f\text{ even}\end{cases}$
A denominator 1 + bᵍ⁽ˣ⁾ with g odd
Halving an even integrand
$\int_{-a}^{a}\frac{h(x)}{1+b^{g(x)}}dx=\int_0^{a}h(x)\,dx$
Periodic integrands
Many periods
$\int_0^{nT}f(x)\,dx=n\int_0^{T}f(x)\,dx$

Watch out for (3)

Check the whole term
→ Odd parts vanish, even parts double
The numerator must be even
→ A denominator 1 + bᵍ⁽ˣ⁾ with g odd
The right period
→ Periodic integrands

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