MathematicsCalculusJEE Main & Advanced

Calculus Formulas

Complete chapter-wise formula sheet for Calculus (Mathematics) covering key equations, parameters, traps, and concept diagrams for JEE Main & Advanced.

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Chapter Directory (26 Chapters)

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Calculus868 Qs

Limits, Continuity & Differentiability

Fundamental algebraic, exponential, logarithmic, and trigonometric limits, continuity conditions, and derivative theorems.

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Calculus552 Qs

Functions & Relations

Domain, range, injective (one-one), surjective (onto), composite functions, inverse functions, and functional equations.

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Calculus590 Qs

Differentiation & Methods

Product rule, quotient rule, chain rule, implicit differentiation, logarithmic differentiation, and higher derivatives.

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Calculus969 Qs

Application of Derivatives (AOD)

Tangents, normals, monotonicity, maxima & minima, Rolle's theorem, and Lagrange Mean Value Theorem (LMVT).

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Calculus765 Qs

Indefinite Integration

Integration by substitution, integration by parts (ILATE), partial fractions, and standard trigonometric integrals.

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Calculus838 Qs

Definite Integration & Properties

Definite integrals, symmetry properties, King's rule, periodic functions, and differentiation under the integral sign.

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Calculus828 Qs

Differential Equations

First-order linear differential equations, integrating factor (IF), homogeneous forms, and variable separable methods.

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Calculus346 Qs

Area Under Curves

Area bounded by curves, lines, parabolas, and standard shortcut formulas.

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Formulas Reference (Calculus)

8 Chapters match

CalculusVery High Yield (3-4 Qs in JEE)868 PYQs

Limits, Continuity & Differentiability

Fundamental algebraic, exponential, logarithmic, and trigonometric limits, continuity conditions, and derivative theorems.

Chapter

Standard Algebraic & Trig Limits

\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan x}{x} = 1, \quad \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}

Fundamental trigonometric and polynomial limits for resolving 0/0 indeterminate forms.

Angle x MUST be measured in radians.

Exponential & Logarithmic Limits

\lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a, \quad \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1

Standard limits for natural exponential and logarithmic expressions.

Useful substitution: lim_{x->0} (ln(1+ax))/x = a.

Exponential 1^∞ Indeterminate Form

\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} g(x)[f(x) - 1]} \quad (\text{when } f(x) \to 1, g(x) \to \infty)

Converts 1^∞ indeterminate power limits into standard exponential products directly.

Very frequent in JEE Main: lim_{x->0} (1 + ax)^(b/x) = e^(ab).

Maclaurin Series Expansions

\sin x = x - \frac{x^3}{6} + \dots, \quad e^x = 1 + x + \frac{x^2}{2} + \dots, \quad \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots

Taylor-Maclaurin series for resolving complex indeterminate forms without repetitive L'Hôpital differentiation.

Keep terms up to the lowest non-vanishing power of x in the denominator.

Continuity Condition at a Point

\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c) \iff \text{LHL} = \text{RHL} = f(c)

A function f(x) is continuous at x = c if and only if both one-sided limits exist and equal the functional value.

Greatest integer [x], fractional part {x}, and signum sgn(x) are classic discontinuity test cases.

Differentiability from First Principles

f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}, \quad \text{LHD} = \text{RHD} \implies \text{Differentiable}

Differentiability implies continuity, but continuity does NOT imply differentiability.

Sharp vertices, cusps, and vertical tangents signify non-differentiable points.

L'Hôpital's Rule for Indeterminate Forms

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \left(\text{Valid strictly for } \frac{0}{0} \text{ and } \frac{\infty}{\infty} \text{ forms}\right)

Differentiate numerator and denominator independently until the indeterminate form is resolved.

Do NOT apply the quotient rule when applying L'Hôpital's rule. Check that the indeterminate form persists before differentiating repeatedly.

Sandwich (Squeeze) Theorem & Limits

g(x) \le f(x) \le h(x) \quad \text{and} \quad \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \implies \lim_{x \to a} f(x) = L

Bounding technique for evaluating limits of functions involving oscillatory factors like [x], {x}, or sin(1/x).

Classic example: lim_{x->0} x² sin(1/x) = 0 because -x² ≤ x² sin(1/x) ≤ x².

Intermediate Value Theorem (IVT)

f(x) \in C[a, b] \text{ and } f(a) \cdot f(b) < 0 \implies \exists c \in (a, b) \text{ such that } f(c) = 0

Fundamental root existence theorem for continuous real-valued functions across opposite-sign endpoints.

Every odd degree real polynomial has at least one real root.

CalculusHigh Yield (2 Qs in JEE)552 PYQs

Functions & Relations

Domain, range, injective (one-one), surjective (onto), composite functions, inverse functions, and functional equations.

Chapter

Domain & Range Rules

\sqrt{f(x)} \implies f(x) \ge 0, \quad \ln(f(x)) \implies f(x) > 0, \quad \frac{1}{f(x)} \implies f(x) \ne 0

Fundamental algebraic domain restrictions for real-valued functions.

Domain of sin⁻¹(f(x)) and cos⁻¹(f(x)) requires -1 ≤ f(x) ≤ 1.

One-One (Injective) Test

f(x_1) = f(x_2) \implies x_1 = x_2 \quad \text{or} \quad f'(x) > 0 \; \forall x \text{ (Strictly Monotonic)}

Horizontal line test: every horizontal line intersects the graph at most once.

Strictly increasing or strictly decreasing continuous functions on an interval are always one-one.

Onto (Surjective) Condition

\text{Range}(f) = \text{Codomain}(f)

Every element in the codomain has at least one pre-image in the domain.

Odd degree polynomial functions with codomain R are always onto R.

Composite & Inverse Functions

(f \circ g)(x) = f(g(x)), \quad f(f^{-1}(x)) = x, \quad (f \circ g)^{-1} = g^{-1} \circ f^{-1}

Function composition and reversal law for inverse functions.

Graph of f⁻¹(x) is the mirror reflection of f(x) across the line y = x.

Standard Functional Equations

f(xy) = f(x) + f(y) \implies f(x) = k\ln x; \quad f(x+y) = f(x)f(y) \implies f(x) = a^x; \quad f(x) + f(1/x) = f(x)f(1/x) \implies f(x) = 1 \pm x^n

Classic functional identities frequently tested in JEE Advanced.

Substitute x = 0, 1, or -1 to find base constants.

Logarithm Fundamental Identities & Base Change

\log_a(xy) = \log_a x + \log_a y, \quad \log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y, \quad \log_b a = \frac{\log_c a}{\log_c b}, \quad a^{\log_b c} = c^{\log_b a}, \quad \log_{b^\beta}(a^\alpha) = \frac{\alpha}{\beta}\log_b a

Exhaustive logarithmic laws and base-change properties for solving exponential and log expressions.

Always ensure base a > 0, a ≠ 1 and argument x > 0 before manipulating equations.

Logarithmic Inequalities & Characteristic

\text{For } \log_a x > \log_a y: \begin{cases} x > y > 0 & \text{if } a > 1 \\ 0 < x < y & \text{if } 0 < a < 1 \end{cases}; \quad x = [x] + \{x\}, \quad \log_{10} N = \text{Char} + \text{Mantissa}

Inequality reversal depending on base a, and number of digits in N is equal to (Characteristic + 1).

Number of leading zeroes before first significant digit after decimal in N (0 < N < 1) is |Characteristic| - 1.

Even, Odd & Periodic Functions

f(-x) = f(x) \iff \text{Even (Symmetric about y-axis)}; \quad f(-x) = -f(x) \iff \text{Odd (Symmetric about origin)}; \quad f(x+T) = f(x) \iff \text{Periodic with Period } T

Symmetry and periodicity properties. Derivative of an even function is odd, and vice-versa.

Period of |sin x|, |cos x|, |tan x| is π. Period of sin(ax + b) is 2π/|a|.

CalculusHigh Yield (2 Qs in JEE)590 PYQs

Differentiation & Methods

Product rule, quotient rule, chain rule, implicit differentiation, logarithmic differentiation, and higher derivatives.

Chapter

Product & Quotient Rules

\frac{d}{dx}(uv) = u'v + uv', \quad \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}

Fundamental derivative rules for algebraic combinations.

Numerator of quotient rule starts with derivative of numerator: u'v.

Chain Rule & Parametric Derivatives

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}, \quad \frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

Derivative of composite functions and second derivative of parametric functions.

Do NOT just divide d²y/dt² by d²x/dt² for second derivative.

Logarithmic Differentiation

y = [u(x)]^{v(x)} \implies \frac{dy}{dx} = [u(x)]^{v(x)} \left[ v'(x)\ln(u(x)) + \frac{v(x)u'(x)}{u(x)} \right]

Differentiating variable-base to variable-exponent functions.

Standard result: d/dx (x^x) = x^x(1 + ln x).

Standard Inverse Trig Derivatives

\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}, \quad \frac{d}{dx}(\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}}

Derivatives of inverse trigonometric functions.

Derivative of co-functions (cos⁻¹x, cot⁻¹x, csc⁻¹x) has identical form with a minus sign.

Leibniz's Rule for n-th Derivative

\frac{d^n}{dx^n}(uv) = \sum_{r=0}^n {}^nC_r \frac{d^{n-r}u}{dx^{n-r}} \frac{d^r v}{dx^r}

Binomial-type expansion for calculating the n-th derivative of a product of two functions.

Set v as the polynomial term whose higher derivatives vanish after a few steps.

Differentiation of Determinants

\frac{d}{dx} \begin{vmatrix} f & g \\ u & v \end{vmatrix} = \begin{vmatrix} f' & g' \\ u & v \end{vmatrix} + \begin{vmatrix} f & g \\ u' & v' \end{vmatrix}

Differentiate one row (or column) at a time, keeping all other rows unchanged, and sum the resulting determinants.

For 3×3 determinants, the result is the sum of 3 determinants corresponding to differentiating row 1, row 2, and row 3.

Implicit Function Differentiation Shortcut

F(x, y) = 0 \implies \frac{dy}{dx} = -\frac{\partial F / \partial x}{\partial F / \partial y}

Fast evaluation using partial derivatives treating y as constant in numerator and x as constant in denominator.

Massive time saver in JEE for finding slopes of tangents to implicit algebraic curves.

CalculusVery High Yield (3-4 Qs in JEE)969 PYQs

Application of Derivatives (AOD)

Tangents, normals, monotonicity, maxima & minima, Rolle's theorem, and Lagrange Mean Value Theorem (LMVT).

Chapter

Tangent and Normal Equations

y - y_1 = m(x - x_1) \quad \left(m = \left.\frac{dy}{dx}\right|_{(x_1, y_1)}\right), \quad \text{Normal: } y - y_1 = -\frac{1}{m}(x - x_1)

Point-slope equations for tangent and perpendicular normal lines to curve y = f(x).

If m = 0, tangent is horizontal (y = y_1). If m -> ∞, tangent is vertical (x = x_1).

Length of Subtangent & Subnormal

\text{Subtangent} = \left|\frac{y}{m}\right|, \quad \text{Subnormal} = |y \cdot m|, \quad \text{Tangent length} = \left|\frac{y}{m}\right|\sqrt{1 + m^2}

Geometric lengths of projections on the x-axis for curve tangents and normals.

Subnormal is proportional to y for parabola y² = 4ax (constant subnormal = 2a).

Monotonicity Criteria

f'(x) \ge 0 \; \forall x \in (a, b) \implies f(x) \text{ Increasing}; \quad f'(x) \le 0 \implies f(x) \text{ Decreasing}

Derivative test for strictly increasing and decreasing functions on open intervals.

f'(x) = 0 at discrete isolated points does NOT destroy strict monotonicity.

Maxima & Minima (First & Second Derivative)

f'(c) = 0: \quad f''(c) < 0 \implies \text{Local Maxima}, \quad f''(c) > 0 \implies \text{Local Minima}

Stationary points and concavity curvature tests for extreme values.

If f''(c) = 0, test higher derivatives: odd order = inflection point, even order = extremum.

Lagrange's Mean Value Theorem (LMVT)

\frac{f(b) - f(a)}{b - a} = f'(c) \quad \text{for at least one } c \in (a, b)

Secant slope across [a, b] equals the tangent derivative slope at some interior point c.

Rolle's Theorem is the special case when f(a) = f(b) (giving f'(c) = 0).

Angle of Intersection of Two Curves & Orthogonality

\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|, \quad \text{Orthogonal Condition: } m_1 m_2 = -1

Acute angle between tangents to curves y = f(x) and y = g(x) at their point of intersection.

Two intersecting confocal conics (e.g. ellipse and hyperbola with same foci) always intersect orthogonally.

Concavity, Convexity & Point of Inflection

f''(x) > 0 \implies \text{Concave Upward } (\cup), \quad f''(x) < 0 \implies \text{Concave Downward } (\cap), \quad f''(c) = 0 \; (\& \text{ changes sign}) \implies \text{Inflection}

Curvature direction and inflection point where tangent crosses the curve.

At a point of inflection, the tangent crosses the curve (e.g. y = x³ at the origin).

Cauchy's Mean Value Theorem (CMVT) & Approximations

\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)} \quad (c \in (a,b)), \quad \Delta y \approx f'(x)\,\Delta x, \quad \frac{\Delta y}{y} \approx \left[\frac{x f'(x)}{f(x)}\right]\frac{\Delta x}{x}

Generalized mean value theorem for two functions, and differential approximation for calculating percentage errors.

Percentage error in y is (% error in x) multiplied by the power index n when y = k x^n.

CalculusVery High Yield (2-3 Qs in JEE)765 PYQs

Indefinite Integration

Integration by substitution, integration by parts (ILATE), partial fractions, and standard trigonometric integrals.

Chapter

Integration by Parts (ILATE Rule)

\int u v \, dx = u \int v \, dx - \int \left( u' \int v \, dx \right) dx

Integration of product of two functions with priority order: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential.

For ∫ ln x dx, treat as ∫ (ln x) · 1 dx (u = ln x, v = 1).

Special Exponential Cancellation Integral

\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C, \quad \int [f(x) + x f'(x)] \, dx = x f(x) + C

Classic JEE pattern where derivative cancels out directly.

Always try to express awkward integrals in the form e^x [f(x) + f'(x)].

Standard Algebraic Quadratic Integrals

\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C, \quad \int \frac{dx}{\sqrt{x^2 \pm a^2}} = \ln|x + \sqrt{x^2 \pm a^2}| + C

Standard formulas for quadratic denominators.

∫ dx / (a² + x²) = (1/a) tan⁻¹(x/a) + C.

Trigonometric Substitution Guidelines

\sqrt{a^2 - x^2} \implies x = a\sin\theta; \quad \sqrt{a^2 + x^2} \implies x = a\tan\theta; \quad \sqrt{x^2 - a^2} \implies x = a\sec\theta

Standard substitutions that eliminate radicals via trigonometric identities.

For √( (a - x)/(a + x) ), substitute x = a cos 2θ.

Three Fundamental Radical Integrals

\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C, \quad \int \sqrt{x^2 \pm a^2} dx = \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\ln|x + \sqrt{x^2 \pm a^2}| + C

Indispensable formulas for area calculations and conic integrals in JEE Advanced.

Memory trick: All three start with (x/2)·[original integrand], followed by (a²/2)·[integral of 1/[original integrand]].

Symmetric Biquadratic Rational Integrals

\int \frac{x^2 \pm 1}{x^4 + kx^2 + 1} dx = \int \frac{1 \pm 1/x^2}{(x \mp 1/x)^2 + (k \pm 2)} dx

Divide numerator and denominator by x² and substitute t = x ∓ 1/x.

If numerator is 1 or x², express as 1/2 · [(x² + 1) ± (x² - 1)].

Linear over Quadratic & Radical Integrals

\int \frac{px + q}{ax^2 + bx + c} dx = \frac{p}{2a} \int \frac{2ax + b}{ax^2 + bx + c} dx + \left(q - \frac{pb}{2a}\right) \int \frac{dx}{ax^2 + bx + c}

Split numerator into derivative of quadratic denominator plus a constant.

The first part integrates directly to (p/2a) ln|ax² + bx + c|, while the second is resolved by completing the square.

Walli's Formula (Definite Integral Reduction)

\int_0^{\pi/2} \sin^n x \, dx = \int_0^{\pi/2} \cos^n x \, dx = \begin{cases} \dfrac{(n-1)!!}{n!!} \cdot \dfrac{\pi}{2} & n \text{ even} \\[6pt] \dfrac{(n-1)!!}{n!!} & n \text{ odd} \end{cases}

Walli's (Wallis) reduction for ∫₀^{π/2} sinⁿx dx. The double-factorial n!! = n·(n-2)·(n-4)···. E.g., ∫₀^{π/2} sin⁴x dx = (3·1)/(4·2) · π/2 = 3π/16.

Quick check: ∫₀^{π/2} sin²x dx = π/4, ∫₀^{π/2} sin³x dx = 2/3, ∫₀^{π/2} sin⁵x dx = 8/15. Walli's eliminates repeated IBP in JEE Advanced.

Reduction Formulas for sin^n x and cos^n x

\int \sin^n x \, dx = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2}x \, dx, \quad \int \cos^n x \, dx = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n} \int \cos^{n-2}x \, dx

Recursive IBP-based formulas to reduce power n by 2 each step until reaching ∫sin x dx or ∫dx. Terminates at base cases n = 0 (→ x) or n = 1 (→ -cos x / sin x).

Also: ∫ sec³x dx = (1/2)[sec x tan x + ln|sec x + tan x|] + C — needed in arc-length and surface-area integrals.

CalculusVery High Yield (3-4 Qs in JEE)838 PYQs

Definite Integration & Properties

Definite integrals, symmetry properties, King's rule, periodic functions, and differentiation under the integral sign.

Chapter

King's Definite Integral Property

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

Transforming x into (a + b - x) leaves the integral value invariant.

Add the new integral to the original integral I to cancel awkward denominators (2I technique).

Symmetric Bounds (Even & Odd Functions)

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

Symmetry test across origin drastically reduces definite integral calculations.

Check odd symmetry immediately for symmetric limits [-a, a].

Periodic Function Integral Rule

\int_0^{nT} f(x) \, dx = n \int_0^T f(x) \, dx, \quad \int_a^{a + T} f(x) \, dx = \int_0^T f(x) \, dx

If f(x + T) = f(x), integrals over multiple periods pull factor n outside.

The integral over any full period T is completely independent of the starting limit a.

Leibniz Differentiation Rule

\frac{d}{dx} \int_{\alpha(x)}^{\beta(x)} f(t) \, dt = f(\beta(x)) \beta'(x) - f(\alpha(x)) \alpha'(x)

Differentiating a definite integral with variable boundary limits with respect to x.

Crucial for applying L'Hôpital's Rule on definite integral indeterminate forms.

Definite Integral as Limit of a Sum

\lim_{n \to \infty} \sum_{r=1}^n \frac{1}{n} f\left(\frac{r}{n}\right) = \int_0^1 f(x) \, dx

Converts infinite Riemann series summations into standard definite integrals (r/n -> x, 1/n -> dx).

Lower bound = lim (r_min/n), Upper bound = lim (r_max/n).

Walli's Reduction Formula

\int_0^{\pi/2} \sin^n x \, dx = \frac{(n-1)(n-3)\dots}{(n)(n-2)\dots} \times K \quad \left(K = \frac{\pi}{2} \text{ if } n \text{ even, } 1 \text{ if odd}\right)

Instant evaluation of definite powers of sine and cosine over [0, π/2].

Saves up to 5 minutes compared to standard reduction integration.

Half-Integral Invariance Property

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

Splits interval [0, 2a] based on reflection symmetry across the midpoint x = a.

Essential for integrating ∫₀^π x f(sin x) dx: first remove x using King's property, then apply half-integral rule.

Definite Integral Estimation & Cauchy-Schwarz Inequality

m(b - a) \le \int_a^b f(x)\,dx \le M(b - a), \quad \left(\int_a^b f(x)g(x)\,dx\right)^2 \le \left(\int_a^b [f(x)]^2\,dx\right)\left(\int_a^b [g(x)]^2\,dx\right)

Bounding bounds for definite integrals where m and M are infimum and supremum of f(x) on [a, b].

Modulus inequality: |∫ f(x) dx| ≤ ∫ |f(x)| dx.

CalculusHigh Yield (2-3 Qs in JEE)828 PYQs

Differential Equations

First-order linear differential equations, integrating factor (IF), homogeneous forms, and variable separable methods.

Chapter

Linear Differential Equation (First Order)

\frac{dy}{dx} + P(x) y = Q(x) \implies y \cdot e^{\int P dx} = \int \left( Q \cdot e^{\int P dx} \right) dx + C

Multiplying by Integrating Factor IF = e^(∫ P dx) makes LHS a total derivative.

If dy/dx looks complicated, check if dx/dy + P(y)x = Q(y) is simpler.

Bernoulli's Differential Equation

\frac{dy}{dx} + P(x) y = Q(x) y^n \implies \frac{1}{1-n} \frac{dz}{dx} + P(x) z = Q(x) \quad (z = y^{1-n})

Reduces non-linear Bernoulli equations into standard first-order linear form.

Divide entire equation by y^n before substituting z = y^(1-n).

Homogeneous Differential Equation

\frac{dy}{dx} = f\left(\frac{y}{x}\right) \implies y = vx, \quad \frac{dy}{dx} = v + x\frac{dv}{dx}

Substitute y = vx to convert homogeneous equations into variable separable form.

Always substitute back v = y/x into the final expression.

Exact Differential Form Substitutions

x \, dy + y \, dx = d(xy), \quad \frac{x \, dy - y \, dx}{x^2} = d\left(\frac{y}{x}\right), \quad \frac{x \, dx + y \, dy}{x^2 + y^2} = d\left(\frac{1}{2}\ln(x^2 + y^2)\right)

Recognizing exact differentials allows one-line integration without lengthy substitutions.

Group x dy and y dx terms together first.

Orthogonal Trajectories

\text{Replace } \frac{dy}{dx} \longrightarrow -\frac{dx}{dy} \text{ in the differential equation}

Finds the family of curves that intersect the given family at right angles everywhere.

Eliminate any arbitrary constants before replacing the derivative.

Clairaut's Equation & Singular Solution

y = px + f(p) \; \left(p = \frac{dy}{dx}\right) \implies \text{General: } y = cx + f(c), \quad \text{Singular: Eliminate } p \text{ via } x + f'(p) = 0

Standard non-linear differential equation whose general solution is a family of straight lines and singular solution is the envelope.

Substitute dy/dx = c directly into the differential equation to get the general solution in 2 seconds.

Linear Fractional Form Reductions

\frac{dy}{dx} = \frac{a_1 x + b_1 y + c_1}{a_2 x + b_2 y + c_2} \implies \begin{cases} \text{If } a_2 = -b_1: \text{ Cross-multiply and integrate directly} \\ \text{If } \frac{a_1}{a_2} = \frac{b_1}{b_2}: \text{ Substitute } t = a_1 x + b_1 y \end{cases}

Shortcuts for non-homogeneous rational fractional linear differential equations.

When a₂ = -b₁, (b₁ y dx + a₂ x dy) becomes exact: d(b₁ x y).

Order, Degree & Newton's Law of Cooling

\text{Degree is undefined if derivatives are within transcendental functions}; \quad \frac{dT}{dt} = -k(T - T_0) \implies T(t) = T_0 + \Delta T_0 \, e^{-kt}

Criteria for polynomial degree in derivatives, and standard differential model for thermal cooling.

dy/dx = sin(y) has degree 1, but sin(dy/dx) = y has undefined degree.

CalculusHigh Yield (1-2 Qs in JEE)346 PYQs

Area Under Curves

Area bounded by curves, lines, parabolas, and standard shortcut formulas.

Chapter

Area Bounded Between Two Curves

\text{Area} = \int_a^b [y_{\text{upper}}(x) - y_{\text{lower}}(x)] \, dx = \int_c^d [x_{\text{right}}(y) - x_{\text{left}}(y)] \, dy

Area enclosed between two intersecting curves.

Find points of intersection first to determine integration limits a and b.

Area Between Parabola and Line (Shortcut)

\text{Area bounded by } y^2 = 4ax \text{ and } y = mx \implies \text{Area} = \frac{8a^2}{3m^3}

Direct closed-form formula saving 5 minutes of definite integration.

Area between y² = 4ax and x² = 4by is (16/3)ab.

Standard Ellipse Area

\text{Area of } \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \text{Area} = \pi a b

Total area enclosed by ellipse with semi-axes a and b.

Area of circle of radius r is special case: π r² (a = b = r).

Area Between Two Intersecting Parabolas

\text{Area bounded by } y^2 = 4ax \text{ and } x^2 = 4by \implies \text{Area} = \frac{16}{3} a b

Legendary JEE shortcut for enclosed area between standard horizontal and vertical parabolas.

For y² = 4x and x² = 4y (a = 1, b = 1), the area is exactly 16/3 sq units.

Area Between Parabola & Latus Rectum

\text{Area bounded by } y^2 = 4ax \text{ and } x = a \implies \text{Area} = \frac{8}{3} a^2

Area bounded by the parabola and the focal chord perpendicular to its principal axis.

Area bounded by y² = 4ax and double ordinate x = x₁ is (4/3) x₁ y₁.

Loop Area of Algebraic Curves

\text{Loop of } a y^2 = x^2 (a - x) \implies \text{Area} = \frac{8}{15} a^2, \quad \text{Arch of Cycloid } = 3\pi a^2

Closed-form definite integration results for parametric and looped algebraic geometries.

Area of leaf of Folium of Descartes x³ + y³ = 3axy is (3/2) a².

Area Enclosed by Sine & Cosine Waves

\int_0^\pi \sin x\,dx = 2, \quad \text{Area between } \sin x \text{ and } \cos x \; \left[\frac{\pi}{4}, \frac{5\pi}{4}\right] = 2\sqrt{2}

Fundamental cycle areas for periodic trigonometric functions.

Area of a single full lobe between consecutive zero crossings of y = sin(kx) is 2/k.

Average & Root Mean Square (RMS) Value of Function

f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx, \quad f_{\text{rms}} = \sqrt{\frac{1}{b - a}\int_a^b [f(x)]^2\,dx}

Continuous integral definitions of the mean value and effective quadratic average over interval [a, b].

For alternating sinusoidal function f(t) = I₀ sin(ωt): f_avg over half-cycle is 2I₀/π ≈ 0.637 I₀; f_rms is I₀/√2 ≈ 0.707 I₀.