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y’=dy/dx ∆=b2-4ac x1= (-b+√∆)/2a x2= (-b+√∆)/2a ||| α*ln|b| = ln|bα| ||| A=ln |eA ||| (x*i)’=i |||∫ ln |x| = x*ln |x| - x ||| ∫ (2v dv) / (v2+1) = ln | v2+1| ||| ∫ xn-1*eb*x^n = (eb*x^n)/n*b ||| ∫ F’(x)/F(x) = ln F(x) ||| ∫ xm*eaxdx= [(xm*eax)/a] – (m/a) * ∫ xm-1* eax dx || ∫ eb*x dx= (eb*x )/b |||

* ∫ xm/(a+bx) = (xm/b*m) – (a/b) * ∫ (xm-1)/a+bx ||∫ xn-1*eb*x^n = [(eb*x^n)/n*b] – 1/(n*b) ||∫ 1/(a + bx)p = -1/[(p-1)*b*(a+bx)p-1] || ∫ 1/(x+a)k=∫(x+a)-k=[(x+a)-k+1]/-k+1

$\,e^1 = e,\ e^{x+y} =e^x \cdot e^y\ .$ $\,\!\, b^x=\left(e^{\ln b}\right)^x=e^{x \ln b}.\,$ ||| $\ln(xy)=\ln(x)+\ln(y)\;$ , $\ln\left( \frac{x}{y}\right) =\ln(x)-\ln(y)\;$ , $\ln e^{x} = x\$ ,

elnx = x dla x > 0, $\int \frac{dx}{x}=\ln |x|+C$ , $\int \frac{f^\prime (x)}{f(x)}dx=\ln |f(x) |+C$ , $\begin{array}{l l}\sin (-x) = -\sin x & \mbox{tg }(-x) = -\mbox{tg } x \\\cos (-x) = \cos x & \mbox{ctg } (-x) = -\mbox{ctg } x\end{array}$ , $\sin (x \pm y) = \sin x \cdot \cos y \pm \cos x \cdot \sin y \,$ , $\cos (x \pm y) = \cos x \cdot \cos y \mp \sin x \cdot \sin y \,$ , $\sin x \pm \sin y = 2 \sin \tfrac {x \pm y} 2 \cdot \cos \tfrac {x \mp y } 2$ $\cos x + \cos y = 2 \cos \tfrac {x + y} 2 \cdot \cos \tfrac {x - y } 2$ , $\cos x - \cos y = -2 \sin \tfrac {x + y} 2 \cdot \sin \tfrac {x - y } 2$ , $\sin 2x = 2 \sin x \cdot \cos x \,$ , $\left.\cos 2x = 2\cos^2x - 1 \right.$ , $\cos x \cdot \cos y = \tfrac{\cos (x - y) + \cos (x + y)} 2$ , $\sin x \cdot \sin y = \tfrac{\cos (x - y) - \cos (x + y)} 2$ , $\sin x \cdot \cos y = \tfrac{\sin (x - y) + \sin (x + y)} 2$

--wymierne--

$\int 1\; \mbox{d}x = x$ |||| $\int x^n \mbox{d}x = \frac{x^{n+1}}{n+1} \qquad (n \neq -1)$ |||| $\int (ax + b)^n \mbox{d}x = \frac{(ax + b)^{n+1}}{a(n + 1)} \qquad\mbox{(dla } n\neq -1\mbox{)}$ |||| $\int \frac1x \mbox{d}x = \ln|x|$

$\int \frac{1}{x^n} \mbox{d}x = \frac{-1}{(n-1)x^{n-1}} \qquad (n \neq 1)$ |||| $\int\frac{\mbox{d}x}{ax + b} = \frac{1}{a}\ln\left|ax + b\right|$ |||| $\int\frac{x\;\mbox{d}x}{ax + b} = \frac{x}{a} - \frac{b}{a^2}\ln\left|ax + b\right|$ |||| $\int\frac{x\;\mbox{d}x}{(ax + b)^2} = \frac{b}{a^2(ax + b)} + \frac{1}{a^2}\ln\left|ax + b\right|$ |||| * $\int\frac{x^2\;\mbox{d}x}{ax + b} = \frac{1}{a^3}\left(\frac{(ax + b)^2}{2} - 2b(ax + b) + b^2\ln\left|ax + b\right|\right)$

$\int\frac{x^2\;\mbox{d}x}{(ax + b)^2} = \frac{1}{a^3}\left(ax + b - 2b\ln\left|ax + b\right| - \frac{b^2}{ax + b}\right)$ |||| $\int\frac{\mbox{d}x}{x^2(ax+b)} = -\frac{1}{bx} + \frac{a}{b^2}\ln\left|\frac{ax+b}{x}\right|$

$\int\frac{x^2\;\mbox{d}x}{(ax + b)^n} = \frac{1}{a^3}\left(-\frac{1}{(n- 3)(ax + b)^{n-3}} + \frac{2b}{(n-2)(a + b)^{n-2}} - \frac{b^2}{(n - 1)(ax + b)^{n-1}}\right) \qquad\mbox{(dla } n\not\in \{1, 2, 3\}\mbox{)}$

$\int\frac{\mbox{d}x}{x(ax + b)} = -\frac{1}{b}\ln\left|\frac{ax+b}{x}\right|$ ||| $\int\frac{\mbox{d}x}{x^2(ax+b)^2} = -a\left(\frac{1}{b^2(ax+b)} + \frac{1}{ab^2x} - \frac{2}{b^3}\ln\left|\frac{ax+b}{x}\right|\right)$ ||| $\int\frac{\mbox{d}x}{x^2+a^2} = \frac{1}{a}\mathrm{arctg}\frac{x}{a}$ $\int\frac{\mbox{d}x}{x^2-a^2} = -\frac{1}{a}\,\mathrm{arctgh}\frac{x}{a} = \frac{1}{2a}\ln\frac{x-a}{x+a} \qquad\mbox{(dla }|x| > |a|\mbox{)}$ $\int\frac{\mbox{d}x}{ax^2+bx+c} = \frac{2}{\sqrt{4ac-b^2}}\mathrm{arctg}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(dla }4ac-b^2>0\mbox{)}$

$\int\frac{\mbox{d}x}{ax^2+bx+c} = \frac{2}{\sqrt{b^2-4ac}}\,\mathrm{artgh}\frac{2ax+b}{\sqrt{b^2-4ac}} = \frac{1}{\sqrt{b^2-4ac}}\ln\left|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right| \qquad\mbox{(dla }4ac-b^2<0\mbox{)}$

$\int\frac{x\;\mbox{d}x}{ax^2+bx+c} = \frac{1}{2a}\ln\left|ax^2+bx+c\right|-\frac{b}{2a}\int\frac{\mbox{d}x}{ax^2+bx+c}$

$\int\frac{mx+n}{ax^2+bx+c}\mbox{d}x = \frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{4ac-b^2}}\mathrm{arctg}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(dla }4ac-b^2>0\mbox{)}$

$\int\frac{mx+n}{ax^2+bx+c}\mbox{d}x = \frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{b^2-4ac}}\,\mathrm{artgh}\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad\mbox{(dla }4ac-b^2<0\mbox{)}$

$\int \frac{\mbox{d}x}{(a+bx^n)^p} = \frac{x}{(n(p-1)a)(a+bx^n)^{p-1}} - \frac{1-np+n}{n(p-1)a} \int \frac{\mbox{d}x}{(a+bx^n)^{p-1}}$

--trygono--

$\int\sin cx\;dx = -\frac{1}{c}\cos cx$ ||| $\int\sin^n cx\;dx = -\frac{\sin^{n-1} cx\cos cx}{nc} + \frac{n-1}{n}\int\sin^{n-2} cx\;dx \qquad\mbox{(dla }n>0\mbox{)}$

$\int x\sin cx\;dx = \frac{\sin cx}{c^2}-\frac{x\cos cx}{c}$ ||| $\int x^n\sin cx\;dx = -\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cx\;dx \qquad\mbox{(dla }n>0\mbox{)}$

$\int\frac{\sin cx}{x^n} dx = -\frac{\sin cx}{(n-1)x^{n-1}} + \frac{c}{n-1}\int\frac{\cos cx}{x^{n-1}} dx$ ||| $\int\frac{dx}{\sin cx} = \frac{1}{c}\ln \left|\tan\frac{cx}{2}\right|$

$\int\frac{dx}{\sin^n cx} = \frac{\cos cx}{c(1-n) \sin^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}cx} \qquad\mbox{(dla }n>1\mbox{)}$ ||| $\int\frac{dx}{1\pm\sin cx} = \frac{1}{c}\tan\left(\frac{cx}{2}\mp\frac{\pi}{4}\right)$

$\int\frac{x\;dx}{1+\sin cx} = \frac{x}{c}\tan\left(\frac{cx}{2} - \frac{\pi}{4}\right)+\frac{2}{c^2}\ln\left|\cos\left(\frac{cx}{2}-\frac{\pi}{4}\right)\right|$ ||| $\int\frac{x\;dx}{1-\sin cx} = \frac{x}{c}\cot\left(\frac{\pi}{4} - \frac{cx}{2}\right)+\frac{2}{c^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{cx}{2}\right)\right|$

$\int\frac{\sin cx\;dx}{1\pm\sin cx} = \pm x+\frac{1}{c}\tan\left(\frac{\pi}{4}\mp\frac{cx}{2}\right)$ ||| $\int\sin c_1x\sin c_2x\;dx = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}-\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(dla }|c_1|\neq|c_2|\mbox{)}$

$\int\cos cx\;dx = \frac{1}{c}\sin cx$

$\int\cos^n cx\;dx = \frac{\cos^{n-1} cx\sin cx}{nc} + \frac{n-1}{n}\int\cos^{n-2} cx\;dx \qquad\mbox{(dla }n>0\mbox{)}$

$\int x\cos cx\;dx = \frac{\cos cx}{c^2} + \frac{x\sin cx}{c}$

$\int x^n\cos cx\;dx = \frac{x^n\sin cx}{c} - \frac{n}{c}\int x^{n-1}\sin cx\;dx$

$\int\frac{\cos cx}{x} dx = \ln|cx|+\sum_{i=1}^\infty (-1)^i\frac{(cx)^{2i}}{2i\cdot(2i)!}$

$\int\frac{\cos cx}{x^n} dx = -\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int\frac{\sin cx}{x^{n-1}} dx \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{dx}{\cos cx} = \frac{1}{c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|$

$\int\frac{dx}{\cos^n cx} = \frac{\sin cx}{c(n-1) cos^{n-1} cx} + \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} cx} \qquad\mbox{(dla }n>1\mbox{)}$

$\int\frac{dx}{1+\cos cx} = \frac{1}{c}\tan\frac{cx}{2}$

$\int\frac{dx}{1-\cos cx} = -\frac{1}{c}\cot\frac{cx}{2}$

$\int\frac{x\;dx}{1+\cos cx} = \frac{x}{c}\tan{cx}{2} + \frac{2}{c^2}\ln\left|\cos\frac{cx}{2}\right|$

$\int\frac{x\;dx}{1-\cos cx} = -\frac{x}{c}\cot{cx}{2}+\frac{2}{c^2}\ln\left|\sin\frac{cx}{2}\right|$

$\int\frac{\cos cx\;dx}{1+\cos cx} = x - \frac{1}{c}\tan\frac{cx}{2}$

$\int\frac{\cos cx\;dx}{1-\cos cx} = -x-\frac{1}{c}\cot\frac{cx}{2}$

$\int\cos c_1x\cos c_2x\;dx = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}+\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(dla }|c_1|\neq|c_2|\mbox{)}$

$\int\tan cx\;dx = -\frac{1}{c}\ln|\cos cx|$

$\int\tan^n cx\;dx = \frac{1}{c(n-1)}\tan^{n-1} cx-\int\tan^{n-2} cx\;dx \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{dx}{\tan cx + 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx + \cos cx|$

$\int\frac{dx}{\tan cx - 1} = -\frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|$

$\int\frac{\tan cx\;dx}{\tan cx + 1} = \frac{x}{2} - \frac{1}{2c}\ln|\sin cx + \cos cx|$

$\int\frac{\tan cx\;dx}{\tan cx - 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|$

$\int\cot cx\;dx = \frac{1}{c}\ln|\sin cx|$

$\int\cot^n cx\;dx = -\frac{1}{c(n-1)}\cot^{n-1} cx - \int\cot^{n-2} cx\;dx \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{dx}{1 + \cot cx} = \int\frac{\tan cx\;dx}{\tan cx+1}$

$\int\frac{dx}{1 - \cot cx} = \int\frac{\tan cx\;dx}{\tan cx-1}$

$\int\frac{dx}{\cos cx\pm\sin cx} = \frac{1}{c\sqrt{2}}\ln\left|\tan\left(\frac{cx}{2}\pm\frac{\pi}{8}\right)\right|$

$\int\frac{dx}{(\cos cx\pm\sin cx)^2} = \frac{1}{2c}\tan\left(cx\mp\frac{\pi}{4}\right)$

$\int\frac{\cos cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} + \frac{1}{2c}\ln\left|\sin cx + \cos cx\right|$

$\int\frac{\cos cx\;dx}{\cos cx - \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|$

$\int\frac{\sin cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx + \cos cx\right|$

$\int\frac{\sin cx\;dx}{\cos cx - \sin cx} = -\frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|$

$\int\frac{\cos cx\;dx}{\sin cx(1+\cos cx)} = -\frac{1}{4c}\tan^2\frac{cx}{2}+\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|$

$\int\frac{\cos cx\;dx}{\sin cx(1+-\cos cx)} = -\frac{1}{4c}\cot^2\frac{cx}{2}-\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|$

$\int\frac{\sin cx\;dx}{\cos cx(1+\sin cx)} = \frac{1}{4c}\cot^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)+\frac{1}{2c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|$

$\int\frac{\sin cx\;dx}{\cos cx(1-\sin cx)} = \frac{1}{4c}\tan^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)-\frac{1}{2c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|$

$\int\sin cx\cos cx\;dx = \frac{1}{2c}\sin^2 cx$

$\int\sin c_1x\cos c_2x\;dx = -\frac{\cos(c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos(c_1-c_2)x}{2(c_1-c_2)} \qquad\mbox{(dla }|c_1|\neq|c_2|\mbox{)}$

$\int\sin^n cx\cos cx\;dx = \frac{1}{c(n+1)}\sin^{n+1} cx \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\sin cx\cos^n cx\;dx = -\frac{1}{c(n+1)}\cos^{n+1} cx \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\sin^n cx\cos^m cx\;dx = -\frac{\sin^{n-1} cx\cos^{m+1} cx}{c(n+m)}+\frac{n-1}{n+m}\int\sin^{n-2} cx\cos^m cx\;dx \qquad\mbox{(dla }m,n>0\mbox{)}$

również: $\int\sin^n cx\cos^m cx\;dx = \frac{\sin^{n+1} cx\cos^{m-1} cx}{c(n+m)} + \frac{m-1}{n+m}\int\sin^n cx\cos^{m-2} cx\;dx \qquad\mbox{(dla }m,n>0\mbox{)}$

$\int\frac{dx}{\sin cx\cos cx} = \frac{1}{c}\ln\left|\tan cx\right|$

$\int\frac{dx}{\sin cx\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx}+\int\frac{dx}{\sin cx\cos^{n-2} cx} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{dx}{\sin^n cx\cos cx} = -\frac{1}{c(n-1)\sin^{n-1} cx}+\int\frac{dx}{\sin^{n-2} cx\cos cx} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{\sin cx\;dx}{\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{\sin^2 cx\;dx}{\cos cx} = -\frac{1}{c}\sin cx+\frac{1}{c}\ln\left|\tan\left(\frac{\pi}{4}+\frac{cx}{2}\right)\right|$

$\int\frac{\sin^2 cx\;dx}{\cos^n cx} = \frac{\sin cx}{c(n-1)\cos^{n-1}cx}-\frac{1}{n-1}\int\frac{dx}{\cos^{n-2}cx} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{\sin^n cx\;dx}{\cos cx} = -\frac{\sin^{n-1} cx}{c(n-1)} + \int\frac{\sin^{n-2} cx\;dx}{\cos cx} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{sin^n cx\;dx}{\cos^m cx} = \frac{\sin^{n+1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-m+2}{m-1}\int\frac{\sin^n cx\;dx}{\cos^{m-2} cx} \qquad\mbox{(dla }m\neq 1\mbox{)}$

również: $\int\frac{sin^n cx\;dx}{\cos^m cx} = -\frac{\sin^{n-1} cx}{c(n-m)\cos^{m-1} cx}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} cx\;dx}{\cos^m cx} \qquad\mbox{(dla }m\neq n\mbox{)}$

również: $\int\frac{sin^n cx\;dx}{\cos^m cx} = \frac{\sin^{n-1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-1}{n-1}\int\frac{\sin^{n-1} cx\;dx}{\cos^{m-2} cx} \qquad\mbox{(dla }m\neq 1\mbox{)}$

$\int\frac{\cos cx\;dx}{\sin^n cx} = -\frac{1}{c(n-1)\sin^{n-1} cx} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{\cos^2 cx\;dx}{\sin cx} = \frac{1}{c}\left(\cos cx+\ln\left|\tan\frac{cx}{2}\right|\right)$

$\int\frac{\cos^2 cx\;dx}{\sin^n cx} = -\frac{1}{n-1}\left(\frac{\cos cx}{c\sin^{n-1} cx)}+\int\frac{dx}{\sin^{n-2} cx}\right) \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n+1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-m-2}{m-1}\int\frac{cos^n cx\;dx}{\sin^{m-2} cx} \qquad\mbox{(dla }m\neq 1\mbox{)}$

również: $\int\frac{\cos^n cx\;dx}{\sin^m cx} = \frac{\cos^{n-1} cx}{c(n-m)\sin^{m-1} cx} + \frac{n-1}{n-m}\int\frac{cos^{n-2} cx\;dx}{\sin^m cx} \qquad\mbox{(dla }m\neq n\mbox{)}$

również: $\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n-1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-1}{m-1}\int\frac{cos^{n-2} cx\;dx}{\sin^{m-2} cx} \qquad\mbox{(dla }m\neq 1\mbox{)}$

--wykładnicze--

$\int e^x dx = e^x$

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

$\int a^x dx = \frac{a^x}{\ln(a)} \qquad\mbox{(dla } a > 0,\mbox{ }a \ne 1\mbox{)}$

$\int a^{cx}\;dx = \frac{1}{c \ln a} a^{cx} \qquad\mbox{(dla } a > 0,\mbox{ }a \ne 1, c \ne 0\mbox{)}$

$\int xe^{cx}\; dx = \frac{e^{cx}}{c^2}(cx-1)$

$\int x^2 e^{cx}\;dx = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right)$

$\int x^n e^{cx}\; dx = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} dx$

$\int\frac{e^{cx}\; dx}{x} = \ln|x| +\sum_{i=1}^\infty\frac{(cx)^i}{i\cdot i!}$

$\int\frac{e^{cx}\; dx}{x^n} = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}}+c\int\frac{e^{cx} dx}{x^{n-1}}\right) \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int e^{cx}\ln x\; dx = \frac{1}{c}\left(e^{cx}\ln|x|-\int\frac{e^{cx} dx}{x}\right)$

$\int e^{cx}\sin bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\sin bx - b\cos bx)$

$\int e^{cx}\cos bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\cos bx + b\sin bx)$

$\int e^{cx}\sin^n x\; dx = \frac{e^{cx}\sin^{n-1} x}{c^2+n^2}(c\sin x-n\cos x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}\sin^{n-2} x\;dx$

$\int e^{cx}\cos^n x\; dx = \frac{e^{cx}\cos^{n-1} x}{c^2+n^2}(c\cos x+n\sin x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}\cos^{n-2} x\;dx$

--logarytm nat—

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

$\int (\ln cx)^n\; dx = x(\ln cx)^n - n\int (\ln cx)^{n-1} dx \qquad\mbox{(dla }n\neq 1\mbox{)}$ ||| $\int \frac{dx}{\ln x} = \ln|\ln x| + \ln x + \sum^\infty_{i=2}\frac{(\ln x)^i}{i\cdot i!}$

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

$\int x^m\ln x\;dx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2}\right) \qquad\mbox{(dla }m\neq 1\mbox{)}$

$\int x^m (\ln x)^n\; dx = \frac{x^{m+1}(\ln x)^n}{m+1} - \frac{n}{m+1}\int x^m (\ln x)^{n-1} dx \qquad\mbox{(dla }m,n\neq 1\mbox{)}$

$\int \frac{(\ln x)^n\; dx}{x} = \frac{(\ln x)^{n+1}}{n+1} \qquad\mbox{(dla }n\neq -1\mbox{)}$

$\int \frac{\ln x\,dx}{x^m} = -\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2 x^{m-1}} \qquad\mbox{(dla }m\neq 1\mbox{)}$

$\int \frac{(\ln x)^n\; dx}{x^m} = -\frac{(\ln x)^n}{(m-1)x^{m-1}} + \frac{n}{m-1}\int\frac{(\ln x)^{n-1} dx}{x^m} \qquad\mbox{(dla }m,n\neq 1\mbox{)}$

$\int \frac{x^m\; dx}{(\ln x)^n} = -\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}} + \frac{m+1}{n-1}\int\frac{x^m dx}{(\ln x)^{n-1}} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int \frac{dx}{x\ln x} = \ln|\ln x|$

$\int \frac{dx}{x^n\ln x} = \ln|\ln x| + \sum^\infty_{i=1} (-1)^i\frac{(n-1)^i(\ln x)^i}{i\cdot i!}$

$\int \frac{dx}{x (\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}} \qquad\mbox{(dla }n\neq 1\mbox{)}$

$\int \sin (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) - \cos (\ln x))$ ||| $\int \cos (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) + \cos (\ln x))$

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