JEE Maths — Calculus Complete Chapter Guide

Chapter Overview & Weightage

Calculus is the single largest contributor to JEE Maths. In JEE Main, you’re looking at 8–10 questions from this chapter alone — that’s roughly 30–35% of the maths section. If you’re aiming for 80+ in maths, calculus is non-negotiable.

JEE Main Weightage (Year-by-Year)

Year	Questions	Marks	Sub-topics that appeared
2024	9	36	Limits (2), AOD (3), Integrals (3), DE (1)
2023	8	32	Limits (1), Derivatives (1), AOD (2), Integrals (3), DE (1)
2022	10	40	Limits (2), AOD (3), Integrals (4), DE (1)
2021	9	36	Limits (2), AOD (2), Integrals (4), DE (1)
2020	8	32	Limits (1), AOD (3), Integrals (3), DE (1)

AOD (Application of Derivatives) and Integrals together account for 60–65% of calculus questions.

JEE Advanced is a different beast — here, calculus questions test conceptual depth. Expect multi-step problems combining AOD with coordinate geometry, or definite integrals with inequality-based bounds.

Key Concepts You Must Know

Ranked by how often they appear in PYQs:

Integrals (Highest Priority)

Standard forms: $\int \frac{dx}{x^2 + a^2}$ , $\int \frac{dx}{\sqrt{a^2 - x^2}}$ , $\int e^x[f(x) + f'(x)]dx$
Integration by parts (ILATE order)
Partial fractions (linear × quadratic denominators)
Definite integral properties — especially King’s property $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$
Area under curves (single curve and between two curves)

Application of Derivatives (High Priority)

Maxima and minima — both local (first/second derivative test) and absolute
Monotonicity — where $f'(x) > 0$ vs $f'(x) < 0$
Tangents and normals
Mean Value Theorem and Rolle’s Theorem (appears in JEE Advanced)

Limits (Medium Priority)

Standard limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ , $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
L’Hôpital’s Rule for $\frac{0}{0}$ and $\frac{\infty}{\infty}$ forms
$1^\infty$ form — this is where most students lose marks

Differential Equations (Lower Priority, but consistent 1 question)

Variable separable
Linear first-order ODEs using integrating factor
Homogeneous DEs

Continuity & Differentiability

Checking continuity at a point (LHL = RHL = f(a))
Differentiability — if $f$ is differentiable, it must be continuous (converse is false)
Common functions: $|x|$ , $[x]$ (greatest integer), $\{x\}$ (fractional part)

Important Formulas

When $\lim_{x \to a} f(x) = 1$ and $\lim_{x \to a} g(x) = \infty$ :

\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} g(x)[f(x)-1]}

When to use: Any time you see a limit of the form $[1 + \text{small thing}]^{\text{big thing}}$ . Check if it’s $1^\infty$ before applying L’Hôpital — L’Hôpital doesn’t work on $1^\infty$ directly.

\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx

Special case (symmetric limits):

\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f \text{ is even} \\ 0 & \text{if } f \text{ is odd} \end{cases}

When to use: When the integral looks unsolvable by substitution. Adding $I$ to itself using King’s property often simplifies everything to a constant.

\int u \cdot v\,dx = u\int v\,dx - \int \left(\frac{du}{dx} \cdot \int v\,dx\right)dx

ILATE priority for $u$ : Inverse trig → Logarithm → Algebraic → Trigonometric → Exponential

Special case: $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$ — memorise this, it’s a direct result.

For $\frac{dy}{dx} + P(x) \cdot y = Q(x)$ :

Integrating Factor $= e^{\int P(x)\,dx}$

Solution: $y \cdot e^{\int P\,dx} = \int Q \cdot e^{\int P\,dx}\,dx + C$

When to use: When you can’t separate variables. Rearrange the DE to match this standard form first.

A = \int_a^b |f(x) - g(x)|\,dx

Find intersection points first to set $a$ and $b$ . Check which curve is on top in $[a, b]$ — the one with larger $y$ values goes in front.

Solved Previous Year Questions

PYQ 1 — Definite Integral (JEE Main 2024, January Shift 2)

Question: Evaluate $\displaystyle\int_0^\pi \frac{x \sin x}{1 + \cos^2 x}\,dx$

Solution:

Let $I = \displaystyle\int_0^\pi \frac{x \sin x}{1 + \cos^2 x}\,dx$

Apply King’s property: replace $x$ with $\pi - x$ :

I = \int_0^\pi \frac{(\pi - x)\sin(\pi - x)}{1 + \cos^2(\pi - x)}\,dx = \int_0^\pi \frac{(\pi - x)\sin x}{1 + \cos^2 x}\,dx

Adding both expressions:

2I = \pi \int_0^\pi \frac{\sin x}{1 + \cos^2 x}\,dx

Now let $t = \cos x$ , so $dt = -\sin x\,dx$ . When $x = 0$ , $t = 1$ ; when $x = \pi$ , $t = -1$ :

2I = \pi \int_1^{-1} \frac{-dt}{1 + t^2} = \pi \int_{-1}^{1} \frac{dt}{1 + t^2} = \pi \left[\tan^{-1} t\right]_{-1}^{1}

2I = \pi\left[\frac{\pi}{4} - \left(-\frac{\pi}{4}\right)\right] = \pi \cdot \frac{\pi}{2} = \frac{\pi^2}{2}

\boxed{I = \frac{\pi^2}{4}}

The pattern here is the classic King’s property trick for integrals with $x$ in the numerator. Whenever you see $\int_0^\pi x \cdot g(\sin x, \cos x)\,dx$ , immediately apply King’s. The $x$ becomes $\pi - x$ , and adding both gives you $\pi$ times a simpler integral.

PYQ 2 — Application of Derivatives (JEE Main 2023, April Shift 1)

Question: The maximum value of $f(x) = \dfrac{\log x}{x}$ for $x > 0$ is:

Solution:

f'(x) = \frac{\frac{1}{x} \cdot x - \log x \cdot 1}{x^2} = \frac{1 - \log x}{x^2}

Set $f'(x) = 0$ : $1 - \log x = 0 \Rightarrow x = e$

Check second derivative (or sign change): for $x < e$ , $f'(x) > 0$ ; for $x > e$ , $f'(x) < 0$ . So $x = e$ is a maximum.

f(e) = \frac{\log e}{e} = \boxed{\frac{1}{e}}

Students often differentiate $\frac{\log x}{x}$ as $\frac{1}{x} \cdot \frac{1}{x} = \frac{1}{x^2}$ , forgetting the quotient rule. Always write it as $\frac{u}{v}$ and apply $\frac{u'v - uv'}{v^2}$ .

PYQ 3 — Differential Equations (JEE Main 2022, June Shift 1)

Question: The solution of $\dfrac{dy}{dx} - \dfrac{y}{x} = 2x^2$ is:

Solution:

This is a linear ODE. Identify $P(x) = -\dfrac{1}{x}$ , $Q(x) = 2x^2$ .

Integrating factor $= e^{\int -\frac{1}{x}\,dx} = e^{-\log x} = \dfrac{1}{x}$

Multiply both sides by $\frac{1}{x}$ :

\frac{1}{x}\frac{dy}{dx} - \frac{y}{x^2} = 2x

Left side is $\dfrac{d}{dx}\left(\dfrac{y}{x}\right)$ . Integrate both sides:

\frac{y}{x} = \int 2x\,dx = x^2 + C

\boxed{y = x^3 + Cx}

Difficulty Distribution

Based on the last 5 years of JEE Main papers:

Difficulty	% of Calculus Questions	Sub-topics
Easy	25%	Standard limits, basic derivatives, variable separable DEs
Medium	55%	Integration by parts, AOD optimization, King’s property
Hard	20%	$1^\infty$ forms, area between curves with tricky bounds, JEE Advanced-style proofs

In JEE Main, the hard 20% usually appears in the integer-type or numerical questions, not the MCQs. The MCQs test whether you know the standard tricks — King’s property, ILATE, the $e^x[f+f']$ formula. Speed matters here. The numerical questions test whether you can set up and execute a multi-step problem cleanly.

Expert Strategy

Week 1 — Build the toolkit: Master all standard integral forms. Write them on a sheet, paste it near your desk. You need these at fingertip speed — pausing to derive $\int \sec x\,dx$ in the exam costs you 2 minutes.

Week 2 — AOD and monotonicity: These problems look scary but follow a tight algorithm: differentiate, find critical points, check sign changes, interpret geometrically. Practice 15–20 problems until the algorithm is automatic.

Week 3 — Definite integrals (the real game): Work through every question that uses King’s property, odd/even functions, and the reduction formula $\int_0^{\pi/2} \sin^m x \cos^n x\,dx$ (Wallis’ formula). These appear in almost every paper.

Topper’s approach: Start with the 2-mark MCQs in calculus and do them first. Calculus questions have high accuracy potential if you know the tricks — don’t waste energy on them in the last 10 minutes. Reserve your difficult numerical questions for when your mind is fresh.

For JEE Advanced: Practice proving results, not just computing them. Questions like “show that $f$ is increasing on $(a,b)$ ” or “find all functions satisfying this functional equation” require you to reason about derivatives, not just calculate.

Spend 40% of your calculus prep time on integration and 35% on AOD. Limits and DEs together need only 25% — they’re more predictable.

Common Traps

Trap 1: Forgetting $|$ absolute value $|$ in area questions

Area is always positive. $\int_a^b f(x)\,dx$ can be negative (when the curve is below the x-axis), but area cannot. Always draw a rough graph, find where $f(x) = 0$ , and split the integral at sign changes. Students who skip the graph lose 4 marks on a problem they otherwise know how to solve.

Trap 2: L’Hôpital on the wrong form

L’Hôpital applies only to $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . For $1^\infty$ , $0 \cdot \infty$ , or $\infty - \infty$ , you must rewrite the expression first. A classic exam trap: $\lim_{x \to 0^+} x \ln x$ . Students apply L’Hôpital directly — wrong form. Rewrite as $\frac{\ln x}{1/x}$ (now it’s $\frac{-\infty}{\infty}$ ), then apply.

Trap 3: Differentiability implies continuity — but not vice versa

$|x|$ is continuous at $x = 0$ but not differentiable. The converse — if differentiable, then continuous — is always true. Examiners love asking “which of these is differentiable at $x = 0$ ?” with options involving $|x|$ , $x|x|$ , $x^2\sin(1/x)$ , and $[x]$ . Know that $x|x|$ and $x^2\sin(1/x)$ (defined as 0 at $x = 0$ ) are differentiable; $|x|$ and $[x]$ are not.

Trap 4: The $+C$ trap in definite integrals

When evaluating a definite integral using substitution, change the limits when you change the variable. Students often substitute $t = g(x)$ but forget to recalculate the bounds in terms of $t$ . This gives a completely wrong numerical answer with no obvious error — a nightmare to debug during the exam.

Trap 5: Second derivative test inconclusive

When $f''(c) = 0$ , the second derivative test tells you nothing. You must go back to the first derivative test (sign change analysis) or check higher derivatives. JEE Advanced occasionally designs problems where $f''(c) = 0$ at a critical point — students who only know the second derivative test will be stuck.