Mathematics — Calculus for the JEE Main Exam

Common failure modes

These are the patterns that cause most candidates to lose marks on this topic. Recognising them in advance is half the work.

• !Forgetting to apply the chain rule in implicit differentiation
• !Confusing L'Hôpital's rule applicability — it requires 0/0 or ∞/∞ indeterminate forms, not all limits
• !Skipping the constant of integration in indefinite integrals
• !Misidentifying the order and degree of a differential equation when the equation contains a radical
• !Using incorrect limits of integration when substituting (forgetting to change bounds)

Study tips

• 1Drill the standard integral table (∫xⁿ, ∫sin x, ∫eˣ, ∫1/x, ∫sec²x) and all inverse-trig integrals until retrieval is instantaneous.
• 2Practice definite integral properties (even/odd symmetry, substitution t = a+b−x) — these appear in JEE Advanced as time-saving shortcuts.
• 3For differential equations, identify type first: separable, homogeneous, linear first-order (integrating factor), Bernoulli. Misclassifying wastes 5+ minutes.
• 4Build the habit of verifying answers to integration problems by differentiation — especially useful for JEE Main where multiple choice allows quick checking.
• 5Solve at least 10 area-between-curves problems; they combine definite integration with coordinate geometry and appear annually.

Sample JEE Main Mathematics — Calculus questions

These sample items mirror the format and difficulty of real JEE Main questions. Practice with thousands more on the free Koydo question bank.

1. 1

   If f(x) = sin(x²), then f'(x) is:

   • Acos(x²)
   • B2x cos(x²)Correct
   • C2x cos(x)
   • Dsin(2x)
   Why this answer?

   Illustrative JEE-style: By the chain rule, d/dx[sin(x²)] = cos(x²) · d/dx(x²) = cos(x²) · 2x = 2x cos(x²).

2. 2

   ∫₀^π sin x dx equals:

   • A0
   • B1
   • C2Correct
   • Dπ
   Why this answer?

   Illustrative JEE-style: ∫₀^π sin x dx = [−cos x]₀^π = −cos(π) + cos(0) = −(−1) + 1 = 2.

3. 3

   The differential equation dy/dx = (x + y)/(x − y) is:

   • ALinear first-order
   • BHomogeneousCorrect
   • CSeparable
   • DBernoulli type
   Why this answer?

   Illustrative JEE-style: Replace x by kx and y by ky: (kx + ky)/(kx − ky) = k(x+y)/k(x−y) = (x+y)/(x−y). The right-hand side is homogeneous of degree 0, so the equation is homogeneous.