Mathematics — Calculus for the JEE Main Exam — Indian candidates

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.
• 6For candidates in India, JEE Main test windows are typically denser in the spring; book test centres in metro cities (Delhi, Mumbai, Bengaluru, Chennai, Kolkata) early to secure preferred dates.

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.

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