Chia sẻ tài liệu Toán THPT Quốc Gia

## 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

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