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# Math formulas:Most popular formulas

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 $$a^2 - b^2 = (a-b)(a+b)$$
 $$a^3 - b^3 = (a-b)\left(a^2 + ab + b^2\right)$$
 $$a^3 + b^3 = (a+b)\left(a^2 - ab + b^2\right)$$
 $$(a + b)^2 = a^2 + 2ab + b^2$$
 $$(a - b)^2 = a^2 - 2ab + b^2$$
 $$x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
 $$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$
 $$A \cup B = \left\{x : x \in A ~~ or ~~ x \in B \right\}$$
 $$A \cap B = \left\{x : x \in A ~~ and ~~ x \in B \right\}$$
 $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$
 $$\sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$$
 $$\sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha$$
 $$\cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta - \sin\alpha \cdot \cos\beta$$
 $$\cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}}$$
 $$\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
 $$\csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}$$
 $$\cos(2\,\alpha) = \cos^2\alpha - \sin^2\alpha$$
 $$\sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha$$
 $$\sin^2x + \cos^2x = 1$$
 $$\tan^2x + 1 = \frac{1}{\cos^2x}$$
 $$y = mx+b$$
 $$C = \pi \cdot d = 2\cdot \pi \cdot r$$
 $$A = r^2\pi$$
 $$y^2 = 2\,p\,x$$
 $$DC^2 = DG \cdot DE$$
 $$DH \cdot DG = DF \cdot DE$$
 \begin{aligned} & \text{if } a \geq b \Longrightarrow y = \frac{b}{a}x \text{ and } y = -\frac{b}{a}x \\ & \text{if } a < b \Longrightarrow y = \frac{a}{b}x \text{ and } y = -\frac{a}{b}x \\ \end{aligned}
 \begin{aligned} & \text{if}~a \geq b \Longrightarrow F_1\left(-\sqrt{a^2+b^2},0\right)~~ F_2\left(\sqrt{a^2+b^2},0\right) \\ & \text{if}~a < b \Longrightarrow F_1\left(0, -\sqrt{a^2+b^2}\right) ~~ F_2\left(0, \sqrt{a^2+b^2}\right) \end{aligned}
 \begin{aligned} x &= \frac{a}{\sin t} \\ y &= \frac{b\,\sin t}{\cos t} \end{aligned}
 $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
 $$\frac{d}{dx} \left( \log_a x \right) = \frac{1}{x\cdot \ln a} , x > 0$$
 $$(f \cdot g)' = f' \cdot g + f \cdot g'$$
 $$\frac{d}{dx} (e^x) = e^x$$
 $$\left( \frac{f}{g} \right)' = \frac{ f'\cdot g - f \cdot g' }{g^2}$$
 $$\left( f \left(g(x) \right) \right)' = f'(g(x)) \cdot g'(x)$$
 $$\frac{d}{dx} (a^x) = a^x \cdot \ln a$$
 $$\frac{d}{dx} (\ln x) = \frac{1}{x} , x > 0$$
 $$\frac{d}{dx} (\arctan x) = \frac{1}{1+x^2}$$
 $$\frac{d}{dx} (\csc x) = - \csc x \cdot \cot x$$
 $$\frac{d}{dx} (\cot x) = -\frac{1}{ \sin^2x }$$
 $$\int \frac{e^{cx}}{x^n} = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}} + c\cdot \int \frac{e^{cx}}{x^{n-1}} dx \right)$$
 $$\int x \cdot e^{cx} = \frac{e^{cx}}{c^2}(cx-1)$$
 $$\int e^{cx}dx = \frac{1}{c}e^{cx}$$
 $$\int \frac{dx}{x\cdot \ln x} = \ln|\ln x|$$
 $$\int \frac{\ln x^n}{x}dx = \frac{\left(\ln x^n \right)^2}{2n},\quad (\text{for } n \ne 0 )$$
 $$\int (\ln x)^2dx = x(\ln x)^2 - 2x\ln x + 2x$$
 $$\int (\ln (cx))^ndx = x(\ln x)^n - n\cdot\int (\ln (cx))^{n-1}dx$$
 $$\int \frac{dx}{\ln x} = \ln|\ln x|+\ln x+\sum\limits_{n=2}^\infty\frac{(\ln x)^i}{i\cdot i!}$$
 $$\int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1} \int \frac{dx}{(\ln x)^{n-1}}$$
 $$\int x^m \cdot \ln xdx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2} \right) \quad ( \text{fot } m\ne1)$$
 $$\int^\infty_0 \frac{dx}{x^2+a^2} = \frac{\pi}{2a}$$
 $$\int^\infty_0 \frac{x^m}{x^n + a^n} = \frac{\pi a^{m + 1 -n}}{n\,\sin[(m+1)\pi/n]}, ~0 < m + 1 < n$$
 $$\int^a_0 \frac{dx}{\sqrt{a^2 - x^2}} = \frac{\pi}{2}$$
 $$\int^a_0 \sqrt{a^2 - x^2}\,dx = \frac{\pi\,a^2}{4}$$
 $$\int^a_0 x^m \left(a^n - x^n\right)^p\,dx = \frac{a^{m+1+np}~\Gamma[(m+1)/n]~\Gamma(p+1) }{n\,\Gamma[(m+1)/n + p +1]}$$
 $$\int^\infty_0 \frac{x^{p-1}\,dx}{1+x} = \frac{\pi}{\sin (p\pi)} , ~ 0 < p < 1$$
 $$\int^\infty_0 \frac{x\,\sin (mx)}{x^2 + a^2} dx = \frac{\pi}{2}e^{-ma}$$
 $$\int^\pi_0 \cos (mx) \cdot \cos (nx)\,dx = \left\{ \begin{array}{l l} 0 & \quad m , n \text{ integers and } m\ne n \\ \pi/2 & \quad m , n \text{ integers and } m = n \end{array} \right.$$
 $$\int^\pi_0 \sin (mx) \cdot \cos (nx)\,dx = \left\{ \begin{array}{l l} 0 & \quad m , n \text{ integers and } m + n \text{ odd} \\ 2m/(m^2 - n^2) & \quad m , n \text{ integers and } m + n \text{ even} \end{array} \right.$$
 $$\int^{\pi/2}_0 \sin^{2m}x\,dx = \int^{\pi/2}_0 \cos^{2m}x\,dx = \frac{1\cdot3\cdot5\dots 2m-1}{2\cdot 4 \cdot 6 \dots 2m} \frac{\pi}{2}$$
 $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$
 $$f(x) = f(a) + f'(a)(x-a) + \frac{f{'}{'}(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n$$
 $$S = \frac{a_1}{1-q}, \quad (\text{for } -1 < q < 1)$$
 $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad -1 < x \leq 1$$
 $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
 $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$
 \begin{aligned} (a + x)^n &= a^n + na^{n-1} + \frac{n(n-1)}{2!} a^{n-2}x^2 + \frac{n(n-1)(n-2)}{3!}a^{n-3}x^3+\cdots \\ &= a^n + { n \choose 1} a^{n-1}x + { n \choose 2} a^{n-2}x^2 + { n \choose 3} a^{n-3}x^3 + \cdots \end{aligned}
 $$\cot x = \frac{1}{x} - \frac{x}{3} - \frac{x^3}{45} - \cdots - \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi$$
 $$\sec x = 1+ \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots + \frac{E_n x^{2n}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}$$
 $$\csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \cdots + \frac{2\left(2^{2n}-1\right)E_n x^{2n}}{(2n)!} \quad 0 < x < \pi$$