JEEClass 12
Calculus Formulas for JEE
Every Calculus formula for JEE Main and Advanced — differentiation rules, standard integrals, definite integral properties, and applications. Calculus makes up ~30% of JEE Maths.
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Differentiation Rules
Power ruled/dx(xⁿ) = nxⁿ⁻¹
Product ruled/dx(uv) = u'v + uv'
Quotient ruled/dx(u/v) = (u'v − uv') / v²
Chain ruled/dx[f(g(x))] = f'(g(x)) · g'(x)
Exponentiald/dx(eˣ) = eˣ, d/dx(aˣ) = aˣ ln(a)
Logarithmicd/dx(ln x) = 1/x, d/dx(log_a x) = 1/(x ln a)
Trigonometricd/dx(sinx) = cosx, d/dx(cosx) = −sinx
Trig (tan, sec)d/dx(tanx) = sec²x, d/dx(secx) = secx tanx
Inverse trigd/dx(sin⁻¹x) = 1/√(1−x²), d/dx(tan⁻¹x) = 1/(1+x²)
Standard Integrals
Power∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
Reciprocal∫ 1/x dx = ln|x| + C
Exponential∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ/ln(a) + C
Trig (sin, cos)∫ sinx dx = −cosx + C, ∫ cosx dx = sinx + C
Trig (sec², csc²)∫ sec²x dx = tanx + C, ∫ csc²x dx = −cotx + C
Trig (sec tan)∫ secx tanx dx = secx + C
Inverse trig type∫ 1/√(1−x²) dx = sin⁻¹x + C
Inverse trig type∫ 1/(1+x²) dx = tan⁻¹x + C
By parts∫ u dv = uv − ∫ v du (ILATE order)
Definite Integral Properties
Limits swap∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx
King's property∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
Even function∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f(−x) = f(x)
Odd function∫₋ₐᵃ f(x)dx = 0 if f(−x) = −f(x)
Periodic∫₀ⁿᵀ f(x)dx = n∫₀ᵀ f(x)dx
Leibniz ruled/dx[∫ₐ⁾ˣᵇ⁾ˣ f(t)dt] = f(b)b' − f(a)a'
Applications
Tangent at (x₁,y₁)y − y₁ = f'(x₁)(x − x₁)
Normal at (x₁,y₁)y − y₁ = −1/f'(x₁) · (x − x₁)
Maxima conditionf'(x) = 0 and f''(x) < 0
Minima conditionf'(x) = 0 and f''(x) > 0
Area under curveA = ∫ₐᵇ f(x) dx
Area between curvesA = ∫ₐᵇ |f(x) − g(x)| dx
Limits (Standard)
sinx/xlim(x→0) sinx/x = 1
(eˣ−1)/xlim(x→0) (eˣ−1)/x = 1
ln(1+x)/xlim(x→0) ln(1+x)/x = 1
(1+1/x)ˣlim(x→∞) (1+1/x)ˣ = e
1∞ formlim f(x)^g(x) = e^{lim g(x)[f(x)−1]}
L'Hôpitallim f/g = lim f'/g' (for 0/0 or ∞/∞)
JEE Tips
- Tip 1:Integration by parts: use ILATE order — Inverse trig > Log > Algebraic > Trig > Exponential
- Tip 2:King's property is the single most useful definite integral trick in JEE
- Tip 3:For maxima/minima, always check endpoints of the domain too