Công thức toán THPT

BẢNG CÔNG THỨC TOÁN THPT

A. ĐẠI SỐ - GIẢI TÍCH

1. CÔNG THỨC LƯỢNG GIÁC

\({\sin ^2}x + {\cos ^2}x = 1\)	\(\tan x \cdot \cot x = 1\)
\(\tan x = \frac{{\sin x}}{{\cos x}}\)	\(1 + {\tan ^2}x = \frac{1}{{{{\cos }^2}x}}\)
\(\cot x = \frac{{\cos x}}{{\sin x}}\)	\(1 + {\cot ^2}x = \frac{1}{{{{\sin }^2}x}}\)
\(\sin 2\alpha = 2\sin \alpha \cdot \cos \alpha \)	\(\sin 3\alpha = 3\sin \alpha - 4{\sin ^3}\alpha \)
\(\cos 2\alpha = {\cos ^2}\alpha - {\sin ^2}\alpha \) \( = 2{\cos ^2}\alpha - 1 = 1 - 2{\sin ^2}\alpha \)	\(\cos 3\alpha = 4{\cos ^3}\alpha - 3\cos \alpha \)
\(\tan 2\alpha = \frac{{2\tan \alpha }}{{1 - {{\tan }^2}\alpha }}\)	\(\tan 3\alpha = \frac{{3\tan \alpha - {{\tan }^3}\alpha }}{{1 - 3{{\tan }^2}\alpha }}\)
\({\sin ^2}\alpha = \frac{{1 - \cos 2\alpha }}{2}\)	\({\tan ^2}\alpha = \frac{{1 - \cos 2\alpha }}{{1 + \cos 2\alpha }}\)
\(\sin (a + b) = \sin a \cdot \cos b + \sin b.\cos a\)	\(\sin (a - b) = \sin a \cdot \cos b - \sin b \cdot \cos a\)
\(\cos (a + b) = \cos a \cdot \cos b - \sin a.\sin b\)	\(\cos (a - b) = \cos a \cdot \cos b + \sin a \cdot \sin b\)
\(\tan (a + b) = \frac{{\tan a + \tan b}}{{1 - \tan a \cdot \tan b}}\)	\(\tan (a - b) = \frac{{\tan a - \tan b}}{{1 + \tan a \cdot \tan b}}\)
\(\cos x + \cos y = 2\cos \frac{{x + y}}{2}\cos \frac{{x - y}}{2}\)	\(\cos x - \cos y = - 2\sin \frac{{x + y}}{2}\sin \frac{{x - y}}{2}\)
\(\sin x + \sin y = 2\sin \frac{{x + y}}{2}\cos \frac{{x - y}}{2}\)	\(\sin x - \sin y = 2\cos \frac{{x + y}}{2}\sin \frac{{x - y}}{2}\)
\(\cos x \cdot \cos y = \frac{1}{2}[\cos (x - y) + \cos (x + y)]\)	\(\sin x\sin y = \frac{1}{2}[\cos (x - y) - \cos (x + y)]\)
\(\sin x\cos y = \frac{1}{2}[\sin (x - y) + \sin (x + y)]\)
\(\tan x + \tan y = \frac{{\sin (x + y)}}{{\cos x\cos y}}\)	\(\tan x - \tan y = \frac{{\sin (x - y)}}{{\cos x\cos y}}\)
\(\cot x + \cot y = \frac{{\sin (x + y)}}{{\sin x\sin y}}\)	\(\cot x - \cot y = \frac{{\sin (x - y)}}{{\sin x\sin y}}\)

2. CÔNG THỨC ĐẠO HÀM

\({\left( {{x^\alpha }} \right)^\prime } = \alpha .{x^{\alpha - 1}}\)	\({\left( {{u^\alpha }} \right)^\prime } = \,\alpha .\,{u^{\alpha - 1}}.u'\)
\({\left( {\frac{1}{x}} \right)^\prime } = \, - \frac{1}{{{x^2}}}\,\,(x \ne 0)\)	\({\left( {\frac{1}{u}} \right)^\prime } = \, - \frac{{u'}}{{{u^2}}}\,\,\left( {u \ne 0} \right)\)
\({\left( {\sqrt x } \right)^\prime } = \frac{1}{{2\sqrt x }}\,\,\left( {x>0} \right)\)	\({\left( {\sqrt u } \right)^\prime } = \frac{{u'}}{{2\sqrt u }}\,\,\left( {u>0} \right)\)
\({\left( {\sin x} \right)^\prime } = \,\cos \,x\)	\({\left( {\sin u} \right)^\prime } = \,u'.\cos \,u\)
\({\left( {\cos x} \right)^\prime } = - \sin x\)	\({\left( {\cos u} \right)^\prime } = \,u'.\cos \,u\)
\({\left( {\tan x} \right)^\prime }\, = \,\frac{1}{{{{\cos }^2}x}}\)	\({\left( {\tan u} \right)^\prime } = \,\frac{{u'}}{{{{\cos }^2}u}}\)
\({\left( {\cot x} \right)^\prime } = - \,\frac{1}{{{{\sin }^2}x}}\)	\({\left( {\cot u} \right)^\prime } = - \,\frac{{u'}}{{{{\sin }^2}u}}\)
\({\left( {{e^x}} \right)^\prime } = \,{e^x}\)	\({\left( {{e^u}} \right)^\prime }\, = \,u'.{e^u}\)
\({\left( {{a^x}} \right)^\prime } = {a^x}.\ln a\)	\({\left( {{a^u}} \right)^\prime } = u'.{a^u}.\ln a\)
\({\left( {\ln \left| x \right|} \right)^\prime } = \frac{1}{x}\)	\({\left( {\ln \left| u \right|} \right)^\prime } = \frac{{u'}}{u}\)
\({\left( {{{\log }_a}\left| x \right|} \right)^\prime } = \frac{1}{{x\ln a}}\)	\({\left( {{{\log }_a}\left| u \right|} \right)^\prime } = \frac{{u'}}{{u.\ln a}}\)

3. CÔNG THỨC NGUYÊN HÀM

\(\int {0dx = C} \)	\(\int {dx = x + C} \)
\(\int {{x^\alpha }dx = \frac{1}{{\alpha + 1}}{x^{\alpha + 1}} + C} \left( {\alpha \ne - 1} \right)\)	\(\int {{{\left( {ax + b} \right)}^\alpha }{\mathop{\rm dx}\nolimits} } = \frac{1}{a}\frac{{{{\left( {ax + b} \right)}^{\alpha + 1}}}}{{\alpha + 1}} + c\,,\alpha \ne - 1\)
\(\int {\frac{1}{{{x^2}}}dx = - \frac{1}{x} + C} \)	\(\int {xdx = \frac{{{x^2}}}{2} + C} \)
\(\int {\frac{1}{x}dx = \ln \left| x \right| + C} \)	\(\int {\frac{{{\mathop{\rm dx}\nolimits} }}{{ax + b}}} = \frac{1}{a}\ln \left| {ax + b} \right| + c\)
\(\int {{e^x}dx = {e^x} + C} \)	\(\int {{e^{ax + b}}dx = \frac{1}{a}{e^{ax + b}} + C} \)
\(\int {{a^x}dx = \frac{{{a^x}}}{{\ln a}} + C} \)	\(\int {{a^{kx + b}}dx = \frac{1}{k}\frac{{{a^{kx + b}}}}{{\ln a}} + C} \)
\(\int {\cos xdx = \sin x + C} \)	\(\int {\cos \left( {ax + b} \right)dx = \frac{1}{a}\sin \left( {ax + b} \right) + C} \)
\(\int {sinxdx = - co{\mathop{\rm s}\nolimits} x + C} \)	\(\int {\sin \left( {ax + b} \right)dx = - \frac{1}{a}\cos \left( {ax + b} \right) + C} \)
\(\int {\tan x.dx\, = - \ln |\cos x| + C} \)	\(\int {\tan \left( {ax + b} \right){\mathop{\rm dx}\nolimits} } = - \frac{1}{a}\ln \left| {\cos \left( {ax + b} \right)} \right| + C\)
\(\int {\cot x.dx\, = \ln |\sin x| + C} \)	\(\int {\cot \left( {ax + b} \right){\mathop{\rm dx}\nolimits} } = \frac{1}{a}\ln \left| {\sin \left( {ax + b} \right)} \right| + C\)
\(\int {\frac{1}{{{{\cos }^2}x}}dx = \tan x + C} \)	\(\int {\frac{1}{{{{\cos }^2}\left( {ax + b} \right)}}dx = \frac{1}{a}\tan \left( {ax + b} \right) + C} \)
\(\int {\frac{1}{{{{\sin }^2}x}}dx = - \cot x + C} \)	\(\int {\frac{1}{{{{\sin }^2}\left( {ax + b} \right)}}dx = - \frac{1}{a}\cot \left( {ax + b} \right) + C} \)
\(\int {\left( {1 + {{\tan }^2}x} \right)dx = \tan x + C} \)	\(\int {\left( {1 + {{\tan }^2}\left( {ax + b} \right)} \right)dx = \frac{1}{a}\tan \left( {ax + b} \right) + C} \)
\(\int {\left( {1 + {{\cot }^2}x} \right)dx = - co{\mathop{\rm t}\nolimits} x + C} \)	\(\int {\left( {1 + {{\cot }^2}\left( {ax + b} \right)} \right)dx = - \frac{1}{a}co{\mathop{\rm t}\nolimits} \left( {ax + b} \right) + C} \)

4. CÔNG THỨC LŨY THỪA

\({a^\alpha } \cdot {a^\beta } = {a^{\alpha + \beta }}\)	\(\frac{{{a^\alpha }}}{{{a^\beta }}} = {a^{\alpha - \beta }};\)
\({({a^\alpha })^\beta } = {a^{\alpha .\beta }}\;;\)	\({(ab)^\alpha } = {a^\alpha } \cdot {b^\alpha };\)
\({\left( {\frac{a}{b}} \right)^\alpha } = \frac{{{a^\alpha }}}{{{b^\alpha }}};\)	\({\left( {\frac{a}{b}} \right)^{ - \alpha }} = {\left( {\frac{b}{a}} \right)^\alpha } \cdot \)
\({a^{ - n}} = \frac{1}{{{a^n}}}\)	\(\sqrt[n]{{{a^m}}} = {a^{\frac{m}{n}}}\)

5. CÔNG THỨC LOGARIT

\({\log _a}1 = 0\)	\({\log _a}a = 1\)
\({\log _a}{N^\alpha } = \alpha .{\log _a}N\)	\({\log _{{a^a}}}N = \frac{1}{\alpha }{\log _a}N\)
\({\log _b}N = \frac{{{{\log }_a}N}}{{{{\log }_a}b}}\)	\({\log _a}N = {\log _a}b \cdot {\log _b}N\)
\({\log _a}b = \frac{1}{{{{\log }_b}a}}\)	\({\log _a}\left( {{N_1}.{N_2}} \right) = {\log _a}{N_1} + {\log _a}{N_2}\)
\({\log _a}\left( {\frac{{{N_1}}}{{{N_2}}}} \right) = {\log _a}{N_1} - {\log _a}{N_2}\)	\({a^{{{\log }_b}c}} = {c^{{{\log }_b}a}}\)

B. HÌNH HỌC