\( f(x) \) | \( \displaystyle \int f(x) dx \) |
---|---|
\( x^n \) | \( \dfrac{x^{n+1}}{n+1} + c\) |
\( \dfrac{1}{x} \) | \( \ln |x| + c \) |
\( e^x \) | \( e^x + c \) |
\( \ln x \) | \( x \ln x - x + c \) |
\( \sin x \) | \( -\cos x + c \) |
\( \cos x \) | \( \sin x + c \) |
\( \tan x \) | \( -\ln |\cos x| + c \) |
\( \cot x \) | \( \ln |\sin x| + c \) |
\( \sec x \) | \( \ln( \sec x + \tan x ) + c \) |
\( \csc x \) | \( \ln(\csc x - \cot x) + c \) |
\( \sinh x \) | \( \cosh x + c \) |
\( \cosh x \) | \( \sinh x + c \) |
\( \tanh x \) | \( \ln( \cosh x) + c \) |
\( \coth x \) | \( \ln( \sinh x) + c \) |
\( \text{sech} \; x \) | \( 2 \tan^{-1}(e^x) + c \) |
\( \text{csch} \; x \) | \( -\ln (\coth x + \text{csch}\; x) + c \) |
\( \dfrac{1}{\sqrt{a^2 - x^2}} \) | \( \sin^{-1} \left(\dfrac{x}{a}\right) + c \; , |x| \lt a \) |
\( \dfrac{1}{\sqrt{a^2 - x^2}} \) | \( - \cos^{-1} \left(\dfrac{x}{a}\right) + c \; , |x| \lt a \) |
\( \dfrac{1}{\sqrt{x^2 + a^2}} \) | \( \ln(x+\sqrt{x^2 + a^2}) + c \) |
\( \dfrac{1}{\sqrt{x^2 - a^2}} \) | \( \ln(x+\sqrt{x^2 - a^2}) + c \) |
\( \dfrac{1}{x^2 + a^2} \) | \( \dfrac{1}{a} \tan^{-1} \left(\dfrac{x}{a} \right) + c \) |
\( \dfrac{1}{x^2 - a^2} \) | \( \dfrac{1}{2 a} \ln \left(\dfrac{x-a}{x+a}\right) + c \) |
\( \dfrac{1}{a^2 - x^2} \) | \( \dfrac{1}{2 a} \ln \left(\dfrac{a+x}{a-x}\right) + c \) |