Derivatives | Integrals: | |
1. | \[\frac{d}{dx}({{x}^{n}})=n.{{x}^{n-1}}\] | \[\int{{{x}^{n}}.dx=\frac{{{x}^{n}}+1}{n+1}+c}\] where \[c=\]integration constant. |
2. | \[\frac{d}{dx}({{e}^{x}})={{e}^{x}}\] | \[\int{{{e}^{x}}.dx={{e}^{x}}+c}\] |
3. | \[\frac{d}{dx}\{\log (x)\}=\frac{1}{x}\] | \[\int{\frac{1}{x}.dx=\log x+c}\] |
4. | \[\frac{d}{dx}({{a}^{x}})={{a}^{x}}.\log a\] | \[\int{{{a}^{x}}.dx=\frac{{{a}^{x}}}{\log a}+c}\] |
5. | \[\frac{d}{dx}(\sin x)=\cos x\] | \[\int{\cos x.dx=\sin x+c}\] |
6. | \[\frac{d}{dx}(cosx)=-\sin x\] | \[\int{\sin xdx=-\cos x+c}\] |
7. | \[\frac{d}{dx}(tanx)={{\sec }^{2}}x\] | \[\int{{{\sec }^{2}}xdx=\tan x+c}\] |
8. | \[\frac{d}{dx}(cotx)=-\cos e{{c}^{2}}x\] | \[\int{\cos e{{c}^{2}}xdx=-\cot x+c}\] |
9. | \[\frac{d}{dx}(secx)=\sec x.\tan x\] | \[\int{\sec x.\tan x.dx=\sec x+c}\] |
10. | \[\frac{d}{dx}(cosecx)=-\cos ecx.\cot x\] | \[\int{\cos ecx.cotx.dx=-\cos ec\,x+c}\] |
11. | \[\frac{d}{dx}(ta{{n}^{-1}}x)=\frac{1}{1+{{x}^{2}}}\] | \[\int{\frac{1}{1+{{x}^{2}}}.dx={{\tan }^{-1}}x+c}\] |
12. | \[\frac{d}{dx}(si{{n}^{-1}}x)=\frac{1}{\sqrt{1-{{x}^{2}}}}\] | \[\int{\frac{1}{\sqrt{1-{{x}^{2}}}}.dx={{\sin }^{-1}}x+c}\] |
13. | \[\frac{d}{dx}(co{{s}^{-1}}x)=\frac{-1}{\sqrt{1-{{x}^{2}}}};\] | \[\int{\frac{1}{\sqrt{1-{{x}^{2}}}}=-{{\cos }^{-1}}x+c}\] |
14. | \[\frac{d}{dx}(ta{{n}^{-1}}x)=\frac{1}{1+{{x}^{2}}};\] | \[\int{\frac{dx}{1+{{x}^{2}}}={{\tan }^{-1}}x+c}\] |
15. | \[\frac{d}{dx}(-co{{t}^{-1}}x)=\frac{1}{1+{{x}^{2}}};\] | \[\int{\frac{1}{1+{{x}^{2}}}dx=-{{\cot }^{-1}}x+c}\] |
16. | \[\frac{d}{dx}(se{{c}^{-1}}x)=\frac{1}{x\sqrt{{{x}^{2}}-1}};\] | \[\int{\frac{1}{x\sqrt{{{x}^{2}}-1}}.dx={{\sec }^{-1}}x+c}\] |
17. | \[\frac{d}{dx}(-cose{{c}^{-1}}x)=\frac{1}{x\sqrt{{{x}^{2}}-1}};\] | \[\int{\frac{dx}{x\sqrt{{{x}^{2}}-1}}=-\cos e{{c}^{-1}}x+c}\] |
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