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This function analyzes the entered mathematical expression and suggests equivalent notations. If the 
button highlights in red after you click it, no equivalents were found.
button highlights in red after you click it, no equivalents were found.Click any of the suggested options to insert it into the input field. Or click the 
button to copy the expression in LaTeX format.
button to copy the expression in LaTeX format.Symbols related to step-by-step computations, such as integrals, limits, or derivatives, are not considered during analysis.
Examples:
\(1\)
\(= \ln\left(e\right)\)
\(= e^{0}\)
\(= \sin\left(\dfrac{\pi}{2}\right)\)
\(= \cos\left(0\right)\)
\(= \tan\left(\dfrac{\pi}{4}\right)\)
\(= \cot\left(\dfrac{\pi}{4}\right)\)
\(= \sec\left(0\right)\)
\(= \csc\left(\dfrac{\pi}{2}\right)\)
\(= 0!\)
\(= 1!\)
\(2+2\)
\(= 4\)
\(= 2^2\)
\(0.2+0.4\)
\(= 0.6\)
\(= \dfrac{3}{5}\)
\(= 60\%\)
\(\dfrac{11}{8}\)
\(= 1.375\)
\(= 137.5\%\)
\(= 1+\dfrac{3}{8}\)
\(-\dfrac{24}{9}\)
\(= -\dfrac{8}{3}\)
\(\approx -2.666666666666666667\)
\(= -3+\dfrac{1}{3}\)
\(\dfrac{2}{4}+\dfrac{6}{5}\)
\(= \dfrac{17}{10}\)
\(= 1.7\)
\(= 170\%\)
\(= 1+\dfrac{7}{10}\)
\(1001\)
\(= 7 \cdot 11 \cdot 13\)
\(12\,345\,678\,901\,234\,567\,890\)
\(=2 \cdot 3^{2} \cdot 5 \cdot 3607 \cdot 3803\)
\(101\)
Prime
* Determining prime numbers up to 1 000 000
\(299\,909\)
Prime
\(\sqrt{529}\)
\(= 23\)
\(\sqrt{68}+\sqrt{17}\)
\(= 3\, \sqrt{17}\)
\(\approx 12.36931687685298\)
\(\dfrac{8}{\sqrt{2^{5}}}\)
\(= \sqrt{2}\)
\(\approx 1.4142135623731\)
\(\dfrac{2^{\frac{5}{2}}}{12}\)
\(= \dfrac{\sqrt{2}}{3}\)
\(\approx 0.47140452079103\)
\(\dfrac{\sqrt{2}}{\sqrt{6}}\)
\(= \dfrac{1}{\sqrt{3}}\)
\(= \dfrac{\sqrt{3}}{3}\)
\(\approx 0.57735026918963\)
\(\sqrt[{3}]{16}\)
\(= 2\, \sqrt[3]{2}\)
\(= \sqrt[3]{2^{4}}\)
\(\approx 2.51984209978975\)
\(10\,\sqrt{8}\)
\(= 20\, \sqrt{2}\)
\(= 5\, \sqrt{2^{5}}\)
\(= 5\, \sqrt{32}\)
\(\approx 28.2842712474619\)
\(\dfrac{1}{\sqrt{2}}\)
\(= \dfrac{\sqrt{2}}{2}\)
\(\approx 0.70710678118655\)
\(= \sin\left(\dfrac{\pi}{4}\right)\)
\(= \cos\left(\dfrac{\pi}{4}\right)\)
\(\sqrt{3}\)
\(\approx 1.73205080756888\)
\(= \tan\left(\dfrac{\pi}{3}\right)\)
\(= \cot\left(\dfrac{\pi}{6}\right)\)
\(\dfrac{1}{2^{\frac{2}{3}}}\)
\(= \dfrac{1}{\sqrt[3]{2^{2}}}\)
\(= \dfrac{1}{\sqrt[3]{4}}\)
\(= \dfrac{\sqrt[3]{2}}{2}\)
\(\approx 0.62996052494744\)
\(\sqrt{2}\)
\(\approx 1.4142135623731\)
\(= \sec\left(\dfrac{\pi}{4}\right)\)
\(= \csc\left(\dfrac{\pi}{4}\right)\)
\(\sqrt{5-2\,\sqrt{6}}\)
\(\approx 0.31783724519578\)
\(= \sqrt{3}-\sqrt{2}\)
\(-\dfrac{2}{3\,\left(\sqrt{3}+\sqrt{2}\right)}\)
\(\approx -0.21189149679719\)
\(= \dfrac{\sqrt{2^{3}}}{3}-\dfrac{2}{\sqrt{3}}\)
\(\sqrt{2}\,\sqrt{3}-\dfrac{\sqrt{2}}{\sqrt{3}}\)
\(= \dfrac{\sqrt{2^{3}}}{\sqrt{3}}\)
\(\approx 1.63299316185545\)
\(= \sqrt{6}-\dfrac{\sqrt{2}}{\sqrt{3}}\)
\(\sin\left(\sqrt{3}\,\sqrt{2}-\dfrac{\sqrt{3}}{\sqrt{2}}\right)\)
\(= \sin\left(\dfrac{\sqrt{3}}{\sqrt{2}}\right)\)
\(\approx 0.94071933374144\)
\(= \sin\left(\sqrt{6}-\dfrac{\sqrt{3}}{\sqrt{2}}\right)\)
\(\sin\left(\dfrac{\pi}{6}\right)\)
\(= \dfrac{1}{2}\)
\(= \sin\left(30\hspace{0.5pt} {^{\circ}}\right)\)
\(= 0.5\)
\(= 50\%\)
\(\sin\left(\dfrac{\pi}{10}\right)\)
\(= \sin\left(18\, {^{\circ}}\right)\)
\(\approx 0.30901699437495\)
\(= \dfrac{\sqrt{5}-1}{4}\)
\(\cos\left(42{^\circ}\right)\)
\(= \cos\left(\dfrac{7\, \pi}{30}\right)\)
\(\approx 0.74314482547739\)
\(= \dfrac{\sqrt{15}-\sqrt{3}+\sqrt{2\, \sqrt{5}+10}}{8}\)
\(\sin\left(\dfrac{113\,\pi}{2}\right)\)
\(= 1\)
\(= \sin\left(\dfrac{\pi}{2}\right)\)
\(= \sin\left(10170\, {^{\circ}}\right)\)
\(= \sin\left(90\, {^{\circ}}\right)\)
\(\tan\left(\dfrac{3\,\pi}{8}\right)\)
\(= \tan\left(67.5\, {^{\circ}}\right)\)
\(\approx 2.4142135623731\)
\(= \sqrt{2}+1\)
\(\sin\left(\dfrac{9\,\pi}{2}-x\right)\)
\(= \cos\left(x\right)\)
\(\sin\left(x+\pi\right)\)
\(= -\sin\left(x\right)\)
\(\sin^{2}\left(2\,x\right)\)
\(= 4\, \sin^{2}\left(x\right)\, \cos^{2}\left(x\right)\)
\(= 1-\cos^{2}\left(2\, x\right)\)
\(= \dfrac{1-\cos\left(4\, x\right)}{2}\)
\(= \dfrac{\tan^{2}\left(2\, x\right)}{\tan^{2}\left(2\, x\right)+1}\)
\(\cos^{3}\left(x\right)\)
\(= \dfrac{\cos\left(3\, x\right)+3\, \cos\left(x\right)}{4}\)
\(\cos\left(x\right)+\cos\left(1\right)\)
\(= 2\, \cos\left(\dfrac{x}{2}+\dfrac{1}{2}\right)\, \cos\left(\dfrac{x}{2}-\dfrac{1}{2}\right)\)
\(\tan\left(x\right)\)
\(= \dfrac{\sin\left(x\right)}{\cos\left(x\right)}\)
\(= \dfrac{\sin\left(2\, x\right)}{\cos\left(2\, x\right)+1}\)
\(= \dfrac{1}{\sin\left(2\, x\right)}-\cot\left(2\, x\right)\)
\(= \csc\left(2\, x\right)-\cot\left(2\, x\right)\)
\(\tan\left(6\,x\right)\)
\(= \dfrac{2\, \tan\left(3\, x\right)}{1-\tan^{2}\left(3\, x\right)}\)
\(\csc\left(x\right)\)
\(= \dfrac{1}{\sin\left(x\right)}\)
\(= \dfrac{\sec\left(x\right)}{\tan\left(x\right)}\)
\(\pi\)
\(= 3.14159\,26535\,89793\,23846\,26433\,83279\,50288\,41971\,69399\,37510\,58209\,74944\,59230\,78164\,06286\,20899\,86280\,34825\,34211\,70679\,82148\ldots\)
\(e\)
\(= 2.71828\,18284\,59045\,23536\,02874\,71352\,66249\,77572\,47093\,69995\,95749\,66967\,62772\,40766\,30353\,54759\,45713\,82178\,52516\,64274\,27466\ldots\)
\(= \sum\limits_{{n}={0}}^{\infty} {\frac{1}{n!}}\)
\(\dfrac{1}{0}\)
Undefined
\(x^{2}+2\,x+1\)
\(= \left(x+1\right)^{2}\)
\(D= b^{2}-4\, a\, c= 0\)
\(x_{1}= -1\)
\(x_{2}= -1\)
\(x^{2}+x+1\)
\(= \left(x+\dfrac{1}{2}\right)^{2}+\dfrac{3}{4}\)
\(D= b^{2}-4\, a\, c= -3\)
\(x_{1}= \dfrac{\sqrt{3}\, i-1}{2}\)
\(x_{2}= \dfrac{-1-\sqrt{3}\, i}{2}\)
\(x^{2}+3\,a\,x+1\)
\(D= b^{2}-4\, a\, c= 9\, a^{2}-4\)
\(x_{1}= \dfrac{\sqrt{9\, a^{2}-4}-3\, a}{2}\)
\( x_{2}= \dfrac{-3\, a-\sqrt{9\, a^{2}-4}}{2}\)
\(\dfrac{3^{2\,n}}{14^{3\,n}}\)
\(= \left(\dfrac{9}{2744}\right)^{n}\)
\(= \dfrac{3^{2\, n}}{2^{3\, n}\, 7^{3\, n}}\)
\(= \dfrac{9^{n}}{2744^{n}}\)
\(8^{x}\)
\(= 2^{3\, x}\)
\(2^{2\,x+3}\)
\(= 8 \cdot 2^{2\, x}\)
\(= 8 \cdot 4^{x}\)
\(= 8+16\, \ln\left(2\right)\, x+16\, \ln^{2}\left(2\right)\, x^{2}+\mathrm{O}\left(x^{3}\right),\; x=0\)
\(2^{2\,x}\cdot 10^{x}\)
\(= 40^{x}\)
\(= 2^{3\, x}\, 5^{x}\)
\(6^{2\,x+\pi}\)
\(= 2^{2\, x}\, 3^{2\, x}\, 6^{\pi}\)
\(= 36^{x}\, 6^{\pi}\)
\(= 6^{\pi}+2\, \ln\left(6\right)\, x\, 6^{\pi}+2\, \ln^{2}\left(6\right)\, x^{2}\, 6^{\pi}+\mathrm{O}\left(x^{3}\right),\; x=0\)
\(\dfrac{6^{x}}{3^{2\,x}-2^{2\,x}}\)
\(= \dfrac{6^{x}}{\left(3^{x}-2^{x}\right)\, \left(2^{x}+3^{x}\right)}\)
\(= -\dfrac{6^{x}}{2^{2\, x}-3^{2\, x}}\)
\(= \dfrac{2^{x}\, 3^{x}}{3^{2\, x}-2^{2\, x}}\)
\(= \dfrac{6^{x}}{9^{x}-4^{x}}\)
\(a\,x^{n}\,b^{n}\)
\(= a\, \left(b\, x\right)^{n}\)
\(\ln\left(100\right)\,x\)
\(= 2\, \ln\left(10\right)\, x\)
\(\ln\left(4\,x\,e^{x}\right)\)
\(= x+\ln\left(x\right)+\ln\left(4\right)\)
\(= \ln\left(x\, e^{x}\right)+\ln\left(4\right)\)
\(\ln\left(x+3\right)\)
\(= \ln\left(1+\dfrac{3}{x}\right)+\ln\left(x\right)\)
\(= \ln\left(\dfrac{x}{3}+1\right)+\ln\left(3\right)\)
\(= \ln\left(3\right)+\dfrac{x}{3}-\dfrac{x^{2}}{18}+\dfrac{x^{3}}{81}+\mathrm{O}\left(x^{4}\right),\; x=0\)
\(\ln\left(x\right)+\ln\left(x+1\right)\)
\(= \ln\left(x^{2}+x\right)\)
\(7\,\ln\left(x\right)-7\,\ln\left(x+1\right)\)
\(= 7\left(\ln\left(x\right)-\ln\left(x+1\right)\right)\)
\(= 7\, \ln\left(\dfrac{x}{x+1}\right)\)
\(= -7\, \ln\left(\dfrac{x+1}{x}\right)\)
\(2\,\ln\left(5\right)+\ln\left(3\right)\)
\(\approx 4.31748811353631\)
\(= \ln\left(75\right)\)
\(\ln\left(12\,a\right)+\ln\left(2\,a\right)-\ln\left(24\,a\right)\)
\(= \ln\left(a\right)\)
\(\ln\left(\dfrac{e^{x}}{2}\right)\)
\(= x-\ln\left(2\right)\)
\(e^{a\,\ln\left(a\right)+b\,\ln\left(b\right)+c\,\ln\left(c\right)}\)
\(= a^{a}\, b^{b}\, c^{c}\)
\(\dfrac{x^{2}+1}{x+1}\)
\(= x-1+\dfrac{2}{x+1}\)
\(\dfrac{1}{x}+\dfrac{1}{x^{2}}\)
\(= \dfrac{x+1}{x^{2}}\)
\(3\,x^{3}+9\,x^{2}+6\,x\)
\(= 3\, x\, \left(x+1\right)\, \left(x+2\right)\)
\(x^{6}+3\)
\(= \left(x^{2}+\sqrt[3]{3}\right)\, \left(x^{2}-\sqrt[3]{9}\, x+\sqrt[3]{3}\right)\, \left(x^{2}+\sqrt[3]{9}\, x+\sqrt[3]{3}\right)\)
\(a\,x^{2}+x^{2}+\sqrt{2}\,x+x\)
\(= \left(a+1\right)\hspace{0.5pt} x^{2}+\left(\sqrt{2}+1\right)\hspace{0.5pt} x\)
\(= x\hspace{0.5pt} \left(a\hspace{0.5pt} x+x+\sqrt{2}+1\right)\)
\(40\,\left(-3\,x+2\,x^{2}\right)^{19}\,\left(2+3\,x\right)^{30}+90\,\left(-3\,x+2\,x^{2}\right)^{20}\,\left(2+3\,x\right)^{29}\)
\(= 40\, \left(3\, x+2\right)^{30}\, \left(2\, x^{2}-3\, x\right)^{19}+90\, \left(3\, x+2\right)^{29}\, \left(2\, x^{2}-3\, x\right)^{20}\)
\(= 10\, x^{19}\, \left(18\, x^{2}-15\, x+8\right)\, \left(2\, x-3\right)^{19}\, \left(3\, x+2\right)^{29}\)
\(= 6\,476\,774\,991\,818\,360\,094\,720\, x^{69}-64\,767\,749\,918\,183\,600\,947\,200\, x^{68}+144\,288\,153\,984\,397\,910\,999\,040\, x^{67}+597\,302\,582\,578\,804\,319\,846\,400\, x^{66}-\ldots-49\,918\,749\,901\,659\,832\,320\, x^{19}\)
\(10\,x^{4}+3\,x^{3}+200171\,x^{2}+60048\,x+220176\)
\(= \left(10\, x^{2}+3\, x+11\right)\, \left(x^{2}+20016\right)\)
\(50\,x^{5}+30\,x^{4}+125\,x^{3}+64\,x^{2}+66\,x+121\)
\(= 10\left(5\, x^{2}+3\, x+11\right)\, \left(x^{3}+\dfrac{3\, x}{10}+\dfrac{11}{10}\right)\)
\(x^{5}+3\,x^{4}+\dfrac{13\,x^{3}}{4}+6\,x^{2}+\dfrac{45\,x}{4}+\dfrac{27}{4}\)
\(= \left(x^{2}+3\, x+\dfrac{9}{4}\right)\, \left(x^{3}+x+3\right)\)
\(= \left(x+\dfrac{3}{2}\right)^{2}\, \left(x^{3}+x+3\right)\)
\(= \dfrac{4\, x^{5}+12\, x^{4}+13\, x^{3}+24\, x^{2}+45\, x+27}{4}\)
\(\sqrt{x^{2}+x^{3}}\)
\(= \left|x\right|\, \sqrt{x+1}\)
\(\left(a^{2}\,y^{2}+x^{2}\right)^{2}-a^{2}\,x^{2}\,y^{2}\)
\(= \left(x^{2}-a\, x\, y+a^{2}\, y^{2}\right)\, \left(x^{2}+a\, x\, y+a^{2}\, y^{2}\right)\)
\(= x^{4}+a^{2}\, x^{2}\, y^{2}+a^{4}\, y^{4}\)
\(= \left(x^{2}+\left(a\, y\right)^{2}\right)^{2}-\left(a\, x\, y\right)^{2}\)
\(\sin^{2}\left(x\right)\,y\,z-2\,\sin\left(x\right)\,y\,z+y\,z+x\,\sin^{2}\left(x\right)\,z+e^{x}\,\sin^{2}\left(x\right)\,z-2\,x\,\sin\left(x\right)\,z-2\,e^{x}\,\sin\left(x\right)\,z+x\,z+e^{x}\,z+x^{2}\,\sin^{2}\left(x\right)\,y+x\,\sin^{2}\left(x\right)\,y+\sin^{2}\left(x\right)\,y-2\,x^{2}\,\sin\left(x\right)\,y-2\,x\,\sin\left(x\right)\,y-2\,\sin\left(x\right)\,y+x^{2}\,y+x\,y+y+x^{3}\,\sin^{2}\left(x\right)+e^{x}\,x^{2}\,\sin^{2}\left(x\right)+x^{2}\,\sin^{2}\left(x\right)+e^{x}\,x\,\sin^{2}\left(x\right)+x\,\sin^{2}\left(x\right)+e^{x}\,\sin^{2}\left(x\right)-2\,x^{3}\,\sin\left(x\right)-2\,e^{x}\,x^{2}\,\sin\left(x\right)-2\,x^{2}\,\sin\left(x\right)-2\,e^{x}\,x\,\sin\left(x\right)-2\,x\,\sin\left(x\right)-2\,e^{x}\,\sin\left(x\right)+x^{3}+e^{x}\,x^{2}+x^{2}+e^{x}\,x+x+e^{x}\)
\(= \left(x^{2}+x+z+1\right)\, \left(\sin\left(x\right)-1\right)^{2}\, \left(x+y+e^{x}\right)\)
\(= x^{3}\, \sin^{2}\left(x\right)-2\, x^{3}\, \sin\left(x\right)+x^{3}+x^{2}\, y\, \sin^{2}\left(x\right)-2\, x^{2}\, y\, \sin\left(x\right)+x^{2}\, y+x^{2}\, \sin^{2}\left(x\right)-2\, x\, \sin\left(x\right)-2\, x^{2}\, \sin\left(x\right)+x^{2}+x\, y\, \sin^{2}\left(x\right)-2\, x\, y\, \sin\left(x\right)+x\, y+x\, z\, \sin^{2}\left(x\right)-2\, x\, z\, \sin\left(x\right)+x\, z+x\, \sin^{2}\left(x\right)+x+y\, z\, \sin^{2}\left(x\right)-2\, y\, z\, \sin\left(x\right)+y\, z+y\, \sin^{2}\left(x\right)-2\, y\, \sin\left(x\right)+y+z\, e^{x}\, \sin^{2}\left(x\right)-2\, z\, e^{x}\, \sin\left(x\right)+z\, e^{x}+x^{2}\, e^{x}\, \sin^{2}\left(x\right)+x\, e^{x}\, \sin^{2}\left(x\right)+e^{x}\, \sin^{2}\left(x\right)-2\, x^{2}\, e^{x}\, \sin\left(x\right)-2\, x\, e^{x}\, \sin\left(x\right)-2\, e^{x}\, \sin\left(x\right)+x^{2}\, e^{x}+x\, e^{x}+e^{x}\)
\(\dfrac{1}{x^{2}+1}+\dfrac{x}{x^{2}+1}+x\)
\(= x+\dfrac{x+1}{x^{2}+1}\)
\(= \dfrac{x\, \left(x^{2}+1\right)+x+1}{x^{2}+1}\)
\(\dfrac{\sqrt{\left(-\frac{2}{3}\right)\,x^{4}+2\,x^{2}}}{\left| x\right|}\)
\(= \dfrac{\sqrt{2\, x^{2}-\frac{2\, x^{4}}{3}}}{\left|x\right|}\)
\(= \dfrac{\sqrt{2}\, \sqrt{3-x^{2}}}{\sqrt{3}}\)
\(\dfrac{x+\sin\left(x\right)}{x\,\sin\left(x\right)}\)
\(= \dfrac{1}{x}+\dfrac{1}{\sin\left(x\right)}\)
\(\dfrac{x^{2}\,y^{3}+b\,y^{2}+3\,a\,x^{2}\,y+3\,a\,b}{y^{2}+3\,a}\)
\(= x^{2}\, y+b\)
\(\dfrac{4\,a\,x^{2}\,y^{2}+4\,a\,y^{2}+b\,x^{2}+a\,x^{2}+b+a+u^{6}+1}{x^{2}+1}\)
\(= \dfrac{u^{6}+\left(4\, a\, x^{2}+4\, a\right)\, y^{2}+a\, x^{2}+b\, x^{2}+a+b+1}{x^{2}+1}\)
\(= 4\, a\, y^{2}+a+b+\dfrac{u^{6}+1}{x^{2}+1}\)
\(\dfrac{x}{\sqrt{3}-1}\)
\(= \left(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}\right)\, x\)
\(\dfrac{1}{a^{2}\,x^{4}-1}\)
\(= \dfrac{1}{\left(a\, x^{2}-1\right)\, \left(a\, x^{2}+1\right)}\)
\(= \dfrac{1}{2\left(a\, x^{2}-1\right)}-\dfrac{1}{2\left(a\, x^{2}+1\right)}\)
\(\dfrac{x+1}{x^{3}+2}\)
\(= \dfrac{x+1}{\left(x+\sqrt[3]{2}\right)\, \left(x^{2}-\sqrt[3]{2}\, x+\sqrt[3]{4}\right)}\)
\(= \dfrac{\left(\frac{1}{3\, \sqrt[3]{2}}-\frac{1}{3\, \sqrt[3]{2^{2}}}\right)\, x+\frac{\sqrt[3]{2^{2}}}{3}+\frac{1}{3}}{x^{2}-\sqrt[3]{2}\, x+\sqrt[3]{2^{2}}}+\dfrac{\frac{1}{3\, \sqrt[3]{2^{2}}}-\frac{1}{3\, \sqrt[3]{2}}}{x+\sqrt[3]{2}}\)
\(\dfrac{\sin\left(x\right)}{\sin^{4}\left(x\right)-1}\)
\(= \dfrac{\sin\left(x\right)}{\left(\sin\left(x\right)-1\right)\, \left(\sin\left(x\right)+1\right)\, \left(\sin^{2}\left(x\right)+1\right)}\)
\(= -\dfrac{\sin\left(x\right)}{2\left(\sin^{2}\left(x\right)+1\right)}-\dfrac{\sin\left(x\right)}{4\left(\sin\left(x\right)+1\right)}+\dfrac{\sin\left(x\right)}{4\left(\sin\left(x\right)-1\right)}\)
\(\dfrac{1}{\sin^{4}\left(x\right)-\cos^{4}\left(x\right)}\)
\(= -\dfrac{1}{\left(\cos\left(x\right)-\sin\left(x\right)\right)\, \left(\sin\left(x\right)+\cos\left(x\right)\right)\, \left(\sin^{2}\left(x\right)+\cos^{2}\left(x\right)\right)}\)
\(= -\dfrac{1}{\left(\cos\left(x\right)-\sin\left(x\right)\right)\, \left(\sin\left(x\right)+\cos\left(x\right)\right)}\)
\(= -\dfrac{1}{\left(\cos^{2}\left(x\right)-\sin^{2}\left(x\right)\right)}\)
\(= -\dfrac{1}{\left(\cos^{4}\left(x\right)-\sin^{4}\left(x\right)\right)}\)
\(= -\dfrac{1}{\cos\left(2\, x\right)}\)
\(\sqrt{x+1}\,\sqrt{x+2}\)
\(= \sqrt{\left(x+1\right)\, \left(x+2\right)}\)
\(= \sqrt{x^{2}+3\, x+2}\)
\(e^{3\,x}\)
\(= \sinh\left(3\, x\right)+\cosh\left(3\, x\right)\)
\(= 1+3\, x+\dfrac{9\, x^{2}}{2}+\mathrm{O}\left(x^{3}\right),\; x=0\)
\(\sin^{2}\left(x\right)+\cos^{2}\left(x\right)+x\)
\(= x+1\)
\(\sqrt{1-\cos^{2}\left(x\right)}\)
\(= \sqrt{-\left(\cos\left(x\right)-1\right)\, \left(\cos\left(x\right)+1\right)}\)
\(= \left|\sin\left(x\right)\right|\)
\(-x\,\sin^{2}\left(x\right)+x\,\cos^{2}\left(x\right)+y\)
\(= x\, \cos\left(2\, x\right)+y\)
\(x+\sin^{2}\left(x\right)\,\cos^{2}\left(x\right)\)
\(= x+\dfrac{\sin^{2}\left(2\, x\right)}{4}\)
\(\sin\left(x\right)\,\cot\left(x\right)\,\tan\left(y\right)\)
\(= \dfrac{\cos\left(x\right)\, \sin\left(y\right)}{\cos\left(y\right)}\)
\(= \cos\left(x\right)\, \tan\left(y\right)\)
\(\dfrac{\sin\left(x\right)}{\cos\left(x\right)}+1\)
\(= \tan\left(x\right)+1\)
\(= \dfrac{\sin\left(x\right)+\cos\left(x\right)}{\cos\left(x\right)}\)
\(\dfrac{1}{\cos\left(x\right)}+1\)
\(= \sec\left(x\right)+1\)
\(= \dfrac{\cos\left(x\right)+1}{\cos\left(x\right)}\)
\(e^{i\,x}\)
\(= \cos\left(x\right)+i\, \sin\left(x\right)\)
\(= 1+i\, x-\dfrac{x^{2}}{2}+\mathrm{O}\left(x^{3}\right),\; x=0\)
\(3\,i\)
\(= 3\left(\cos\left(\dfrac{\pi}{2}\right)+i\, \sin\left(\dfrac{\pi}{2}\right)\right)\)
\(= 3\, e^{\frac{i\, \pi}{2}}\)
\(\left(-1\right)^{\frac{1}{3}}\)
\(= -1\)
\(= \dfrac{1}{2}+\dfrac{\sqrt{3}\, i}{2}\;\;\left(\mathbb{C}\right)\)
\(= \dfrac{1+\sqrt{3}\, i}{2}\)
\(k = 0\colon \dfrac{1}{2}+\dfrac{\sqrt{3}\, i}{2}\)
\(k = 1\colon -1\)
\(k = 2\colon \dfrac{1}{2}-\dfrac{\sqrt{3}\, i}{2}\)
\(\dfrac{1+2\,i}{3+4\,i}\)
\(= \dfrac{1}{2}-\dfrac{1}{2\left(3+4\, i\right)}\)
\(= \dfrac{11}{25}+\dfrac{2\, i}{25}\)
\(= \dfrac{\left(3-4\, i\right)\, \left(1+2\, i\right)}{25}\)
\(\sqrt{1+i}\)
\(= \dfrac{\sqrt{\sqrt{2}+1}}{\sqrt{2}}+\dfrac{i}{\sqrt{2}\hspace{0.5pt} \sqrt{\sqrt{2}+1}}\)
\(k = 0\colon \sqrt[4]{2}\, \left(\cos\left(\dfrac{\pi}{8}\right)+i\, \sin\left(\dfrac{\pi}{8}\right)\right)\)
\(k = 1\colon \sqrt[4]{2}\, \left(\cos\left(\dfrac{9\, \pi}{8}\right)+i\, \sin\left(\dfrac{9\, \pi}{8}\right)\right)\)
\(\sin\left(i+1\right)\)
\(= \sin\left(1\right)\, \cosh\left(1\right)+\cos\left(1\right)\, \sinh\left(1\right)\, i\)
\(\ln\left(1+i\right)\)
\(= \dfrac{\ln\left(2\right)}{2}+\dfrac{i\, \pi}{4}\)
\(\cos\left(i\,x\right)\)
\(= \cosh\left(x\right)\)
\(= \frac{\frac{1}{e^{x}}+e^{x}}{2}\)
\(= 1+\dfrac{x^{2}}{2}+\dfrac{x^{4}}{24}+\mathrm{O}\left(x^{6}\right),\; x=0\)
\(i\,\sin\left(x\right)\)
\(= \sinh\left(i\, x\right)\)
\(= \frac{e^{i\, x}-\frac{1}{e^{i\, x}}}{2}\)
\(3\,\cos\left(x\right)+3\,i\,\sin\left(x\right)\)
\(= 3\left(\cos\left(x\right)+i\, \sin\left(x\right)\right)\)
\(= 3\, e^{i\, x}\)
\(\csc\left(i\,a\,x\right)\)
\(= -\dfrac{i}{\sinh\left(a\, x\right)}\)
\(= \dfrac{1}{\sin\left(i\, a\, x\right)}\)
\(= -i\, \mathrm{csch}\left(a\, x\right)\)
\(= -\dfrac{2\, i}{e^{a\, x}-\frac{1}{e^{a\, x}}}\)
\(3\,\left(e^{5\,x}+\dfrac{1}{e^{5\,x}}\right)\)
\(= 3\, e^{5\, x}+\dfrac{3}{e^{5\, x}}\)
\(= 6\, \cosh\left(5\, x\right)\)
\(\cos\left(2\,\mathrm{atan}\left(x\right)\right)\)
\(= \dfrac{1}{x^{2}+1}-\dfrac{x^{2}}{x^{2}+1}\)
\(2\,\arcsin\left(x\right)+2\,\arccos\left(x\right)\)
\(\pi\)
\(\sinh^{3}\left(2\,x\right)\)
\(= 8\, \sinh^{3}\left(x\right)\, \cosh^{3}\left(x\right)\)
\(= \dfrac{\sinh\left(6\, x\right)-3\, \sinh\left(2\, x\right)}{4}\)
\(\cosh\left(x+2\right)\)
\(= \sinh\left(2\right)\, \sinh\left(x\right)+\cosh\left(2\right)\, \cosh\left(x\right)\)
\(= \dfrac{e^{x+2}+e^{-x-2}}{2}\)
\(c\,\sinh\left(x\right)+c\,\sinh\left(1\right)\)
\(= c\, \left(\sinh\left(x\right)+\sinh\left(1\right)\right)\)
\(= 2\, c\, \sinh\left(\dfrac{x}{2}+\dfrac{1}{2}\right)\, \cosh\left(\dfrac{x}{2}-\dfrac{1}{2}\right)\)
\(1+\cosh\left(2\,x\right)\)
\(= 2\, \cosh^{2}\left(x\right)\)
\(1-\tanh^{2}\left(x\right)\)
\(= -\left(\tanh\left(x\right)-1\right)\, \left(\tanh\left(x\right)+1\right)\)
\(= 1-\dfrac{\sinh^{2}\left(x\right)}{\cosh^{2}\left(x\right)}\)
\(= \dfrac{1}{\cosh^{2}\left(x\right)}\)
\(= \mathrm{sech}^{2}\left(x\right)\)
\(\mathrm{asinh}\left(x\right)\)
\(= \ln\left(x+\sqrt{x^{2}+1}\right)\)
\(= x-\dfrac{x^{3}}{6}+\dfrac{3\, x^{5}}{40}+\mathrm{O}\left(x^{7}\right),\; x=0\)
\(\phi\)
\(= \dfrac{\sqrt{5}+1}{2}\)
\(= 1+\dfrac{1}{\phi}\)
\(= 2\, \cos\left(\dfrac{\pi}{5}\right)\)
\(= 2\, \sin\left(\dfrac{3\, \pi}{10}\right)\)
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