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Integrals
Question | Answer |
---|---|
∫u dv du | uv + ∫ v du |
∫1/x dx | ln|x| +C |
∫a^x dx | a^x / (ln(a)) +C |
∫sin(x) dx | -cos(x) +C |
∫cos(x) dx | sin(x) +C |
∫tan(x) dx | ln|sec(x)| +C |
∫cot(x) dx | ln|sin(x)| +C |
∫sec(x) dx | ln|sec(x) + tan(x)| +C = ln|tan(u/2 + pi/4)| +C |
∫csc(x) dx | ln|tan(x/2)| +C = -ln|csc(x) + cot(x)| +C |
∫sec^2(x) dx | tan(x) +C |
∫csc^2(x) dx | -cot(x) +C |
∫tan^2(x) dx | tan(x)-x +C |
∫cot^2(x) dx | -cot(x)-x +C |
∫sin^2(x) dx | x/2-(sin^2(x)/4) + C |
∫cos^2(x) dx | ½(x + ¼sin(2x)) + C |
∫sec(x)tan(x) dx | sec(x) +C |
∫csc(x)cot(x) dx | -csc(x) +C |
∫1/(x^2 + a^2) dx | tan^-1(x/a)/a +C |
∫1/(x^2 - a^2) dx | ln((x-a)/(x+a))/2a +C |
∫1/sqrt(a^2 - x^2) dx | sin^-1(x/a) +C |
∫1/sqrt(x^2 + a^2) dx | ln(x + sqrt(x^2 + a^2)) +C |
∫1/sqrt(x^2 - a^2) dx | ln(x + sqrt(x^2 - a^2)) +C |
∫1/(x sqrt(x^2 - a^2)) dx | 1/(a sec|x/a|) +C |
∫1/(x sqrt(x^2 + a^2)) dx | -ln((a + sqrt(x^2 + a^2))/x)/a |
∫1/(x sqrt(x^2 - a^2)) dx | -ln((a + sqrt(x^2 - a^2))/x)/a |
∫x e^x dx | x e^x - e^x +C |
∫e^x dx | e^x +C |
∫ln(x) dx | x ln(x) - x + C |