NDA Maths · Definite Integration

Absolute Value, Piecewise and Greatest-Integer Integrals

When the integrand changes formula across the interval — an absolute value, a piecewise rule, or a greatest-integer function — split the integral at every break-point and integrate each piece on its own.

All Definite Integration notes NDA Maths strategy

Why this matters

17 PYQs built on one habit: never integrate across a break-point. For |f(x)| the break-points are where f changes sign; for greatest-integer functions they are where the integer part jumps. Split, integrate each piece, add. Get the break-points right and these are reliable marks.

Concept 1 of 2

Integrating absolute-value and piecewise functions

Intuition

An absolute value hides a sign change. To integrate

|f(x)|

, find where

f

is zero, split the interval there, and on each piece replace

|f|

+f

-f

according to its sign. The same split-and-integrate idea handles any piecewise rule.

Definition

The method for $\int_a^b |f(x)|\,dx$ :

Find the zeros of $f$ inside $[a,b]$ — these are the sign changes.
On each subinterval, $|f| = f$ where $f\ge 0$ and $|f| = -f$ where $f< 0$ .
Integrate each piece and add. Because $|f|\ge 0$ , the answer is the total (unsigned) area, so $\int_a^b|f|\,dx \ge \big|\int_a^b f\,dx\big|$ .

Worked example

Evaluate

\displaystyle\int_2^8 |x-5|\,dx

$x-5$ is zero at $x=5$ : negative on $[2,5]$ , positive on $[5,8]$ .
$\int_2^5 (5-x)\,dx + \int_5^8 (x-5)\,dx$ .
Each is a triangle of base 3, height 3: $\tfrac12\cdot3\cdot3 = \tfrac92$ each.

Answer:

\frac92+\frac92 = 9

Practice this conceptself-check · 3 quick reps

Try it yourself

Evaluate

\displaystyle\int_0^{\pi/2} |\sin x - \cos x|\,dx

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
Where do you split $\int|x^2-1|\,dx$ ?
2.
$\int_{-2}^{-1}\frac{x}{|x|}\,dx = ?$
3.
If $\int_a^b f = p$ and $\int_a^b|f| = q$ , which is larger?

From the bank · past-year question

Example 1Definite IntegrationMODERATE

\displaystyle\int_0^{\pi/2}|\sin x-\cos x|\,dx

is equal to

[Q82 · Apr · 2019]

Find the zeros first — don't drop the absolute value

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

(no bars) gives a SIGNED area, but

\int_{-1}^{1}|x^2-1|\,dx

needs the sign of

x^2-1

on each piece. Forgetting the bars (or the split) gives the wrong, signed value.

Drill 9 more on integrating absolute-value and piecewise functions

Concept 2 of 2

Integrating greatest-integer (floor) functions

Intuition

The greatest-integer function ⌊x⌋ is constant between consecutive integers and jumps by 1 at each integer. To integrate it (or anything built from it), split the interval at every point where the integer part changes, replace ⌊·⌋ by its constant value on each piece, and add.

Definition

Working with ⌊·⌋ under an integral:

⌊x⌋ is constant on $[n, n+1)$ with value $n$ ; split at each integer.
For ⌊x²⌋ or ⌊√x⌋, split where the INSIDE crosses an integer (e.g. ⌊x²⌋ jumps at $x=1,\sqrt2,\sqrt3,\dots$ ).
Useful identity: $\lfloor x\rfloor + \lfloor -x\rfloor = -1$ for non-integer $x$ , so $\int_a^b(\lfloor x\rfloor+\lfloor -x\rfloor)\,dx = a-b$ .
The fractional part $x-\lfloor x\rfloor$ has $\int_n^{n+1}(x-\lfloor x\rfloor)\,dx = \tfrac12$ .

Worked example

Evaluate

\displaystyle\int_0^{1.5} \lfloor x\rfloor\,dx

On $[0,1)$ : $\lfloor x\rfloor = 0$ . On $[1,1.5]$ : $\lfloor x\rfloor = 1$ .
$\int_0^1 0\,dx + \int_1^{1.5} 1\,dx = 0 + 0.5$ .

Answer:

0.5

Practice this conceptself-check · 3 quick reps

Try it yourself

Evaluate

\displaystyle\int_0^{\sqrt2} \lfloor x^2\rfloor\,dx

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
Value of $\lfloor x\rfloor$ on $[-1,0)$ ?
2.
$\int_n^{n+1}(x-\lfloor x\rfloor)\,dx = ?$
3.
$\lfloor x\rfloor + \lfloor -x\rfloor$ for non-integer x?

From the bank · past-year question

Example 2Definite IntegrationMODERATE

What is

\int_{0}^{\sqrt{2}}[x^2]\,dx

equal to (where [.] is greatest integer function)?

[Q82 · Apr · 2018]

⌊x⌋ on [−1, 0) is −1, not 0

The floor of a negative non-integer rounds DOWN:

\lfloor -0.3\rfloor = -1

. A common slip is using

\lfloor x\rfloor=0

[-1,0)

— it is

-1

there, which flips the sign of the contribution.

Drill 6 more on integrating greatest-integer (floor) functions

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.

Watch out for (2)

Find the zeros first — don't drop the absolute value
→ Integrating absolute-value and piecewise functions
⌊x⌋ on [−1, 0) is −1, not 0
→ Integrating greatest-integer (floor) functions

Mastery check — 5 interleaved questions

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

Example 1Definite IntegrationMODERATE

What is

\int_{e^{-1}}^{e^2}\left|\dfrac{\ln x}{x}\right|\,dx

equal to?

[Q100 · Apr · 2017]

Example 2Definite IntegrationMODERATE

Let

f(x) = [\sqrt{x}]

, where

[\cdot]

is the greatest integer function.

What is

\int_{\sqrt{2}}^{\sqrt{3}} f(x)\,dx

equal to?

[Q99 · Apr · 2025]

Example 3Definite IntegrationEASY

Let

p=\displaystyle\int_a^b f(x)\,dx

and

q=\displaystyle\int_a^b|f(x)|\,dx

. If

f(x)=e^{-x}

, then which one of the following is correct?

[Q77 · Apr · 2024]

Example 4Definite IntegrationMODERATE

What is

\int_a^b [x]\,dx + \int_a^b [-x]\,dx

equal to, where

[.]

is the greatest integer function?

[Q81 · Sep · 2018]

Example 5Definite IntegrationEASY

Consider the following for the items that follow: Let

I=\int_{a}^{b}\dfrac{|x|}{x}\,dx

a<b

What is

I

equal to when

a<b<0

[Q85 · Apr · 2023]

Drill every past-year question on this subtopic

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

Drill the 17 questions