Supreme Formula Sheet for Calculus Flashcards

Vocabulary flashcards covering differential calculus rules, standard indefinite integrals, advanced integration techniques, and definite integral properties.

Product Rule (Leibniz Rule)

$$\frac{d}{dx}(uv) = u'v + uv'$$

Quotient Rule

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2}$$

Chain Rule

For $y = f(g(x))$, $\frac{dy}{dx} = f'(g(x)) g'(x)$

Power Rule (Differentiation)

$$\frac{d}{dx}(x^n) = n x^{n-1}$$

Derivative of Exponential $a^x$

$$\frac{d}{dx}(a^x) = a^x \ln(a)$$

Derivative of Logarithm $\log_a(x)$

$$\frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}$$

Derivative of Inverse Sine

$$\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$$

Derivative of Inverse Tangent

$$\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$$

Derivative of Inverse Secant

$$\frac{d}{dx}(\sec^{-1}(x)) = \frac{1}{|x|\sqrt{x^2-1}}$$

Parametric Differentiation

If $x = f(t)$ and $y = g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ and $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$

Logarithmic Differentiation

Technique used for functions of the form $[f(x)]^{g(x)}$ by taking $\ln(y) = g(x) \ln(f(x))$ before deriving.

Implicit Differentiation

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

Power Law (Integration)

$\int x^n dx = \frac{x^{n+1}}{n+1} + c$ where $n \neq -1$

Integral of $\tan(x)$

$$\int \tan(x) dx = \ln|\sec(x)| + c = -\ln|\cos(x)| + c$$

Integral of $\sec(x)$

$$\int \sec(x) dx = \ln|\sec(x) + \tan(x)| + c = \ln\left|\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right| + c$$

Integral of $\csc(x)$

$$\int \csc(x) dx = \ln|\csc(x) - \cot(x)| + c = \ln\left|\tan\left(\frac{x}{2}\right)\right| + c$$

Standard Integral: $\int \frac{1}{x^2+a^2} dx$

$$\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + c$$

Standard Integral: $\int \frac{1}{x^2-a^2} dx$

$$\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + c$$

Standard Integral: $\int \frac{1}{a^2-x^2} dx$

$$\frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + c$$

Standard Integral: $\int \frac{1}{\sqrt{a^2-x^2}} dx$

$$\sin^{-1}\left(\frac{x}{a}\right) + c$$

Standard Integral: $\int \sqrt{a^2-x^2} dx$

$$\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + c$$

Standard Integral: $\int \sqrt{x^2+a^2} dx$

$$\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln|x + \sqrt{x^2+a^2}| + c$$

Integration by Parts (IBP)

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

LIATE / ILATE Rule

Selection order for the first function $u(x)$ in Integration by Parts: Inverse Trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.

Classic JEE Trick Integral

$$\int e^x (f(x) + f'(x)) dx = e^x f(x) + c$$

Euler's Trig Product Integral (Sine)

$$\int e^{ax} \sin(bx) dx = \frac{e^{ax}}{a^2+b^2} (a \sin(bx) - b \cos(bx)) + c$$

Euler's Trig Product Integral (Cosine)

$$\int e^{ax} \cos(bx) dx = \frac{e^{ax}}{a^2+b^2} (a \cos(bx) + b \sin(bx)) + c$$

King's Property

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

Queen's Property

$\int_0^{2a} f(x) dx = 2 \int_0^a f(x) dx$ if $f(2a-x) = f(x)$, and $0$ if $f(2a-x) = -f(x)$

Odd/Even Property (Definite Integration)

$\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx$ if $f(-x) = f(x)$ (even), and $0$ if $f(-x) = -f(x)$ (odd).

Leibniz Rule (Differentiation Under Integral Sign)

If $I(x) = \int_{g(x)}^{h(x)} f(t) dt$, then $\frac{dI}{dx} = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x)$

Wallis Formula

Shortcut for $\int_0^{\pi/2} \sin^m(x) \cos^n(x) dx = \frac{[(m-1)(m-3)\dots][(n-1)(n-3)\dots]}{(m+n)(m+n-2)\dots} \times K$, where $K = \frac{\pi}{2}$ if both $m$ and $n$ are even, else $K = 1$.