 $$\int (A\textcolor{blue}{f(x)} + B\textcolor{brown}{g(x)} dx = A \int \textcolor{blue}{f(x)}dx = B \int \textcolor{brown}{g(x)} dx$$ $$\int \textcolor{ blue}{f'(} \textcolor{ brown}{g(x)} \textcolor{ blue}{)} \textcolor{ brown}{g'(x)}dx = \textcolor{ blue}{f(} \textcolor{ brown}{g(x)} \textcolor{ blue}{)} + C$$ $$\int \textcolor{blue}{U(x)} \textcolor{brown}{dV(x)} = \textcolor{ blue}{U(x)} \textcolor{ brown}{V(x)} - \int \textcolor{ brown}{V(x)} \textcolor{ blue}{dU(x)}$$ $$\int_ \textcolor{ orange}{a}^ \textcolor{ magenta}{b} \textcolor{ blue}{f'(x)}dx = \textcolor{ blue}{f(} \textcolor{ magenta}{b} \textcolor{ blue}{)}- \textcolor{ blue}{f(} \textcolor{ orange}{a} \textcolor{ blue}{)}$$ $$\displaystyle \frac {d} {dx} \int_ \textcolor{ orange}{a}^ \textcolor{ magenta}{x} \textcolor{ blue}{f(t)}dt = \textcolor{ blue}{f(} \textcolor{ magenta}{x} \textcolor{ blue}{)}$$
 $$\int \textcolor{orange}{x}^\textcolor{magenta}{n}dx = \displaystyle \frac {1} {\textcolor{magenta}{n}+1} \textcolor{orange}{x}^{\textcolor{magenta}{n}+1} + C \,$$ , if $$\textcolor{magenta}{n} \ne -1$$ $$\int \displaystyle \frac {dx}{\textcolor{orange}{x}} = \ln|\textcolor{orange}{x}| + C$$ $$\int e^\textcolor{orange}{x}dx = e^\textcolor{orange}{x} + C$$ $$\int \textcolor{magenta}{a}^\textcolor{orange}{x} = \displaystyle \frac {\textcolor{magenta}{a}^\textcolor{orange}{x}} {\ln(\textcolor{magenta}{a})} + C$$ $$\int \sin(\textcolor{orange}{x})dx = -\cos(\textcolor{orange}{x}) + C$$ $$\int \cos(\textcolor{orange}{x})dx = \sin(\textcolor{orange}{x}) + C$$ $$\int \sec^2(\textcolor{orange}{x})dx = \tan(\textcolor{orange}{x}) + C$$ $$\int \csc^2(\textcolor{orange}{x})dx = -\cot(\textcolor{orange}{x}) + C$$ $$\int \sec(\textcolor{orange}{x})\tan(\textcolor{orange}{x})dx = \sec(\textcolor{orange}{x}) + C$$ $$\int \csc( \textcolor{ orange}{x})\cot( \textcolor{ orange}{x})dx = -\csc( \textcolor{ orange}{x}) + C$$ $$\int \tan( \textcolor{orange}{x})dx = \ln|\sec( \textcolor{ orange}{x})| + C$$ $$\int \cot( \textcolor{ orange}{x})dx = \ln|\sin( \textcolor{ orange}{x})| + C$$ $$\int \sec( \textcolor{ orange}{x})dx = \ln|\sec( \textcolor{ orange}{x}) + \tan( \textcolor{ orange}{x})| + C$$ $$\int \csc( \textcolor{ orange}{x})dx = \ln|\csc( \textcolor{ orange}{x}) - \cot( \textcolor{ orange}{x})| + C$$ $$\int \displaystyle \frac {dx} {\sqrt{ \textcolor{ magenta}{a}^2- \textcolor{ orange}{x}^2}} = \sin^{-1} \left( \displaystyle \frac { \textcolor{ orange}{x}} { \textcolor{ magenta}{a}} \right)+ C \, ( \textcolor{ magenta}{a} > 0,\, | \textcolor{ orange}{x}| < \textcolor{ magenta}{a})$$ $$\int \displaystyle \frac {dx} { \textcolor{ magenta}{a}^2+ \textcolor{ orange}{x}^2} = \displaystyle \frac {1} { \textcolor{ magenta}{a}} \tan^{-1} \left( \displaystyle \frac { \textcolor{ orange}{x}} { \textcolor{ magenta}{a}} \right) + C \, ( \textcolor{ magenta}{a} > 0)$$ $$\int \displaystyle \frac {dx} { \textcolor{ magenta}{a}^2- \textcolor{ orange}{x}^2} = \displaystyle \frac {1} { 2 \textcolor{ magenta}{a}} \ln \left| \displaystyle \frac { \textcolor{ orange}{x}+ \textcolor{ magenta}{a}} { \textcolor{ orange}{x}- \textcolor{ magenta}{a}} \right| + C \, ( \textcolor{ magenta}{a}>0)$$ $$\int \displaystyle \frac {dx} { \textcolor{ orange}{x} \sqrt{ \textcolor{ orange}{x}^2- \textcolor{ magenta}{a}^2}} = \displaystyle \frac {1} { \textcolor{ magenta}{a}} \sec^{-1} \left| \displaystyle \frac { \textcolor{ orange}{x}} { \textcolor{ magenta}{a}} \right| + C \, ( \textcolor{ magenta}{a}>0,\, | \textcolor{ orange}{x}|> \textcolor{ magenta}{a})$$