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Common Integrals

Indefinite Integral Method of substitution \( \int f\left(g(x)\right)\cdot g'(x) dx = \int f(u) du \) Integration by parts \( \int f(x) \cdot g'(x)dx = f(x) \cdot g(x) – \int g(x) \cdot f'(x)dx \) Integrals of Rational and Irrational Functions \( \int x^n dx = \frac{x^{n+1}}{n+1} + C , n \ne 1 \) \( \int \frac{1}{x} dx = \ln|x| […]

Higher-order Derivatives Formulas

Definitions and properties Second derivative \( f” = \frac{d}{dx} \left(\frac{dy}{dx}\right) – \frac{d^2y}{dx^2} \) Higher-Order derivative \( f^{(n)} = \left( f^{(n-1)} \right)’ \) \( \left(f \, \pm \, g\right)^{(n)} = f^{(n)} \pm ~g^{(n)} \) Leibniz’s Formulas \( (f \cdot g)” = f” \cdot g + 2 \cdot f’\cdot g’ + f \cdot g” \) \( (f \cdot g)”’ = f”’ […]

Circle Formulas

Equation of a circle In an x−y coordinate system, the circle with center (a,b) and radius r is the set of all points (x,y) such that: \( (x-a)^2 + (y-b)^2 =r^2 \) Circle centered at the origin: \( x^2 + y^2 = r^2 \) Parametric equations \( \begin{aligned} x &= a + r\,\cos t \\ y&= b + […]

Triangles in Two Dimensions

Area of the triangle The area of the triangle formed by the three lines: \( \begin{aligned} A_1x + B_1y + C_1 &= 0 \\ A_2x + B_2y + C_2 &= 0 \\ A_3x + B_3y + C_3 &= 0 \end{aligned} \) is given by \( A = \frac{\begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 […]

Common Derivatives Formulas

Basic Properties of Derivatives \( \left(c \cdot f(x)\right)’ = c \cdot f'(x) \) \( \left(f \pm g \right)’ = f’ \pm g’ \) Product rule \( (f \cdot g)’ = f’ \cdot g + f \cdot g’ \) Quotient rule \( \left( \frac{f}{g} \right)’ = \frac{ f’\cdot g – f \cdot g’ }{g^2} \) Chain rule \( \left( f […]

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