(i)\(\,\frac{d}{dx}\left( \frac{{{x}^{n+1}}}{n+1} \right)={{x}^{n}},n\ne -1\,\Rightarrow \int {{x}^{n}}dx=\frac{{{x}^{n+1}}}{n+1}+C,\,n\ne 1\).
(ii)\(\frac{d}{dx}\left( \log x \right)=\frac{1}{x}\,\Rightarrow \int \frac{1}{x}dx=\log \left| x \right|+C\).
(iii)\(\frac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}\,\Rightarrow {{e}^{x}}dx={{e}^{x}}+C\).
(iv)\(\frac{d}{dx}\left( -\cos x \right)=\sin x\,\Rightarrow \int \sin xdx=-\cos x+C\).
(v)\(\frac{d}{dx}\left( \sin x \right)=\cos x\,\Rightarrow \int \cos xdx=\sin x+C\).
(vi)\(\frac{d}{dx}\left( \tan x \right)={{\sec }^{2}}x\,\Rightarrow \int {{\sec }^{2}}xdx=\tan x+C\).
(vii)\(\frac{d}{dx}\left( -\cot x \right)=\cos e{{c}^{2}}x\,\Rightarrow \int \cos e{{c}^{2}}xdx=-\cot x+C\).
(viii)\(\frac{d}{dx}\left( \sec x \right)=\sec x\tan x\,\Rightarrow \int \sec x\tan xdx=\sec +C\).
(ix)\(\frac{d}{dx}\left( -co\sec x \right)=\cos ecx\cot x\,\Rightarrow \int \cos ecx\cot xdx=-\cos ecx+C\).
(x)\(\frac{d}{dx}\left( \log \sin x \right)=\cot x\,\Rightarrow \cot xdx=\log \left| \sin x \right|+C\).
(xi)\(\frac{d}{dx}\left( -\log \cos x \right)=\tan x\,\Rightarrow \tan xdx=-\log \left| \cos x \right|+C\).
(xii)\(\frac{d}{dx}\left( \log \left( \sec x+\tan x \right) \right)=\sec x\,\Rightarrow \int \sec xdx=\log \left| \sec x+\tan x \right|+C\).
(xiii)\(\frac{d}{dx}\left( \log \left( \cos ecx-\cot x \right) \right)=\cos ecx\,\Rightarrow \int \cos ecxdx=\log \left| \cos ecx-\cot x \right|+C\).
(xiv)\(\frac{d}{dx}\left( {{\sin }^{-1}}\frac{x}{a} \right)=\frac{1}{\sqrt{{{a}^{1}}-{{x}^{2}}}}\,\Rightarrow \int \frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=si{{n}^{-1}}\left( \frac{x}{a} \right)+C\).
(xv)\(\frac{d}{dx}\left( {{\cos }^{-1}}\frac{x}{a} \right)=-\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\,\Rightarrow \int -\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx={{\cos }^{-1}}\left( \frac{x}{a} \right)+C\).
(xvi)\(\frac{d}{dx}\left( \frac{1}{a}{{\tan }^{-1}}\frac{x}{a} \right)=\frac{1}{{{a}^{2}}+{{x}^{2}}}\,\Rightarrow \int \frac{1}{{{a}^{2}}+{{x}^{2}}}dx={{\tan }^{-1}}\left( \frac{x}{a} \right)+C\).
(xvii)\(\frac{d}{dx}\left( \frac{1}{a}{{\cot }^{-1}}\frac{x}{a} \right)=-\frac{1}{{{a}^{2}}+{{x}^{2}}}\,\Rightarrow \int -\frac{1}{{{a}^{2}}+{{x}^{2}}}dx=\frac{1}{a}{{\cot }^{-1}}\left( \frac{x}{a} \right)+C\).
(xviii)\(\frac{d}{dx}\left( \frac{1}{a}\cos e{{c}^{-1}}\frac{x}{a} \right)=-\frac{1}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}\,\Rightarrow \int -\frac{1}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\frac{1}{a}{{\sec }^{-1}}\left( \frac{x}{a} \right)+C\).