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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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