User:Alexlaker

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\int \frac{1}{x^3}\,\, dx = \int x^{-3}\,\, dx = -\frac{1}{2x^2} + c
\int \frac{1}{x^2}\,\, dx = \int x^{-2}\,\, dx = -\frac{1}{x} + c
\int \frac{1}{x}\,\, dx = \int x^{-1}\,\, dx = \ln x + c
\int dx = x + c
\int x\,\, dx = \frac{x^2}{2} + c
\int x^2\,\, dx = \frac{x^3}{3} + c
\int x^3\,\, dx = \frac{x^4}{4} + c
\int x^n\,\, dx = \frac{x^{n+1}}{n+1} + c
\sum_{r=1}^n r = \frac{1}{2}n(n+1)
\sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1)
\sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2
α2 + β2 = (α + β)2 − 2αβ
\frac{d}{dx}\,x^n = nx^{n-1}

\int_a^b f(x)\,\, dx \approx \frac{1}{2}h\Big[y_0+y_n + 2(y_1 + y_2 +...+y_{n-1})\Big]
Where h = \bigg(\frac{b-a}{n}\bigg)

(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + ... + \binom{n}{r}a^{n-r}b^r + ... + \binom{n}{n-1}ab^{n-1} + b^n = \sum_{r=0}^n \binom{n}{r}a^{n-r}b^r
Where \binom{n}{r} = \frac{n!}{r!\,(n-r)!}

\int_p^q f(x)\,\, dx = \lim_{a \rightarrow p^{+}} \int_a^q f(x)\,\, dx

\begin{align} \int_3^{\infty} \frac{1}{\sqrt{x}} \,\, dx & = \lim_{a \rightarrow \infty} \int_3^a \frac{1}{\sqrt{x}} \,\, dx \\ & = \lim_{a \rightarrow \infty} \bigg[2\sqrt{x}\bigg]_3^a \\ & = \lim_{a \rightarrow \infty} 2\sqrt{a} - 2\sqrt{3} \end{align}

\int \sin x \,\, dx = -\cos x + c

\frac{d}{dx}\,\cos x = -\sin x

\frac{d}{dx}\,\cos x = -\sin x

\int \cos x \,\, dx = \sin x + c

-mkv^n = m\frac{dv}{dt}

-kv^n = \frac{dv}{dt}

-k = \frac{1}{v^n}\frac{dv}{dt}

\int -k \,\, dt = \int \frac{1}{v^n} \,\, dv

-kt = -\frac{1}{v^{n-1}(n-1)} + c

t=0,\,v=U

c = \frac{1}{U^{n-1}(n-1)}

\frac{1}{v^{n-1}(n-1)} = \frac{1}{U^{n-1}(n-1)} + \frac{ktU^{n-1}(n-1)}{U^{n-1}(n-1)}

v^{n-1} = \frac{U^{n-1}}{1 + ktU^{n-1}(n-1)}