All Maths Formulas

$$a^3 + b^3 = (a+b)\left(a^2 - ab + b^2\right)$$

$$a^3 - b^3 = (a-b)\left(a^2 + ab + b^2\right)$$

$$a^2 - b^2 = (a-b)(a+b)$$

$$(a - b)^2 = a^2 - 2ab + b^2$$

$$(a + b)^2 = a^2 + 2ab + b^2$$

$$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$

$$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$

$$x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

$$a^4 - b^4 = (a-b)(a+b)\left(a^2 + b^2\right)$$

$$a^5 - b^5 = (a-b)\left(a^4 + a^3b + a^2b^2 + ab^3 + b^4\right)$$

$$\sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$$

$$\cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}}$$

$$\cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta - \sin\alpha \cdot \cos\beta$$

$$\sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha$$

$$\sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha$$

$$\cos(2\,\alpha) = \cos^2\alpha - \sin^2\alpha$$

$$\csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}$$

$$\sin^2x + \cos^2x = 1$$

$$\tan^2x + 1 = \frac{1}{\cos^2x}$$

$$\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

$$y = mx+b$$

$$y^2 = 2\,p\,x$$

$$C = \pi \cdot d = 2\cdot \pi \cdot r$$

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

$$\begin{aligned} x &= \frac{a}{\sin t} \\ y &= \frac{b\,\sin t}{\cos t} \end{aligned}$$

$$e = \frac{\sqrt{a^2-b^2}}{a}$$

$$\frac{x_0\,x}{a^2} + \frac{y_0\,y}{b^2} = 1$$

$$\begin{aligned} & \text{if}~a \geq b \Longrightarrow F_1\left(-\sqrt{a^2-b^2},0\right)~~ F_2\left(\sqrt{a^2-b^2},0\right) \\ & \text{if}~a < b \Longrightarrow F_1\left(0, -\sqrt{b^2-a^2}\right) ~~ F_2\left(0, \sqrt{b^2-a^2}\right) \end{aligned}$$

$$\begin{aligned} & \text{if}~a \geq b \Longrightarrow F_1\left(-\sqrt{a^2+b^2},0\right)~~ F_2\left(\sqrt{a^2+b^2},0\right) \\ & \text{if}~a < b \Longrightarrow F_1\left(0, -\sqrt{a^2+b^2}\right) ~~ F_2\left(0, \sqrt{a^2+b^2}\right) \end{aligned}$$

$$A = r^2\pi$$

$$\frac{d}{dx} \left( \log_a x \right) = \frac{1}{x\cdot \ln a} , x > 0$$

$$(f \cdot g)' = f' \cdot g + f \cdot g'$$

$$\frac{d}{dx} (\ln x) = \frac{1}{x} , x > 0$$

$$\left( f \left(g(x) \right) \right)' = f'(g(x)) \cdot g'(x)$$

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

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

$$\left( \frac{f}{g} \right)' = \frac{ f'\cdot g - f \cdot g' }{g^2}$$

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

$$\frac{d}{dx} (\csc x) = - \csc x \cdot \cot x$$

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

$$\int x \cdot e^{cx} = \frac{e^{cx}}{c^2}(cx-1)$$

$$\int e^{cx}dx = \frac{1}{c}e^{cx}$$

$$\int \frac{e^{cx}}{x^n} = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}} + c\cdot \int \frac{e^{cx}}{x^{n-1}} dx \right)$$

$$\int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1} \int \frac{dx}{(\ln x)^{n-1}}$$

$$\int x^m \cdot \ln xdx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2} \right) \quad ( \text{fot } m\ne1)$$

$$\int x^m \cdot (\ln x)^ndx = \frac{x^{m+1}(\ln x)^n}{m+1} - \frac{n}{m+1}\int x^m(\ln x)^{n-1}dx \quad (\text{for } m \ne 1)$$

$$\int \frac{dx}{\ln x} = \ln|\ln x|+\ln x+\sum\limits_{n=2}^\infty\frac{(\ln x)^i}{i\cdot i!}$$

$$\int (\ln x)^2dx = x(\ln x)^2 - 2x\ln x + 2x$$

$$\int \ln(ax+b)dx = x\ln(ax+b) - x + \frac{b}{a}\ln(ax + b)$$

$$\int \frac{(\ln x)^n}{x}dx = \frac{(\ln x)^{n+1}}{n+1}, \quad(\text{for } n\ne 1)$$

$$\int^a_0 \sqrt{a^2 - x^2}\,dx = \frac{\pi\,a^2}{4}$$

$$\int^\infty_0 \frac{dx}{x^2+a^2} = \frac{\pi}{2a}$$

$$\int^a_0 \frac{dx}{\sqrt{a^2 - x^2}} = \frac{\pi}{2}$$

$$\int^a_0 x^m \left(a^n - x^n\right)^p\,dx = \frac{a^{m+1+np}~\Gamma[(m+1)/n]~\Gamma(p+1) }{n\,\Gamma[(m+1)/n + p +1]}$$

$$\int^\infty_0 \frac{x^m}{x^n + a^n} = \frac{\pi a^{m + 1 -n}}{n\,\sin[(m+1)\pi/n]}, ~0 < m + 1 < n$$

$$\int^\infty_0 \frac{x^{p-1}\,dx}{1+x} = \frac{\pi}{\sin (p\pi)} , ~ 0 < p < 1$$

$$\int^\infty_0 \sin (ax^n)\,dx = \frac{1}{na^{1/n}} \Gamma(1/n)\,\sin \frac{\pi}{2n} , ~~ n > 1$$

$$\int^{\pi}_0 \frac{\cos (mx)\,dx}{1 - 2a\,\cos x + a^2} = \frac{\pi a^m}{1 - a^2}, ~~ a^2 < 1$$

$$\int^{\pi}_0 \frac{x\,\sin x\,dx}{1 - 2a\,\cos x + a^2} = \left\{ \begin{array}{l l l} \frac{\pi}{a} ln(1+a) & |a| < 1 \\ \pi \, ln(1 + \frac{1}{a}) & |a| > 1 \\ \end{array} \right.$$

$$\int^\infty_0 \cos (ax^n)\,dx = \frac{1}{na^{1/n}} \Gamma(1/n)\,\cos \frac{\pi}{2n} , ~~ n > 1$$

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

$$a^x = 1 + x\,\ln a + \frac{(x\,\ln a)^2}{2!} + \frac{(x\,\ln a)^3}{3!} + \cdots$$

$$\begin{aligned} (a + x)^n &= a^n + na^{n-1} + \frac{n(n-1)}{2!} a^{n-2}x^2 + \frac{n(n-1)(n-2)}{3!}a^{n-3}x^3+\cdots \\ &= a^n + { n \choose 1} a^{n-1}x + { n \choose 2} a^{n-2}x^2 + { n \choose 3} a^{n-3}x^3 + \cdots \end{aligned}$$

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$

$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$

$$\csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \cdots + \frac{2\left(2^{2n}-1\right)E_n x^{2n}}{(2n)!} \quad 0 < x < \pi$$

$$\sec x = 1+ \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots + \frac{E_n x^{2n}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}$$

$$\cot x = \frac{1}{x} - \frac{x}{3} - \frac{x^3}{45} - \cdots - \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi$$

$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n$$

$$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad -1 < x \leq 1$$