User:R36/Stuff About Integrals

Differentiation formulas (Assume $f(x)$ and $g(x)$ exist, $a$ and $c$ are constants)			Integration formulas
1. $y=c\cdot x^{n}\rightarrow {\frac {dy}{dx}}=c\cdot nx^{n-1}$	2. $y=f(x)\cdot g(x)\rightarrow {\frac {dy}{dx}}=f(x)\cdot g'(x)+g(x)\cdot f'(x)$		20. $\int u^{n}\,du={\frac {u^{n+1}}{n+1}}+c~({\mbox{if}}~n\neq 1)$	21. $\int {\frac {du}{u}}=\ln |u|+c$
3. $y={\frac {f(x)}{g(x)}}\rightarrow {\frac {dy}{dx}}={\frac {g(x)\cdot f'(x)-f(x)\cdot g'(x)}{[g(x)]^{2}}}$	4. $y=[f(x)]^{n}\rightarrow {\frac {dy}{dx}}=n\cdot [f(x)]^{n-1}\cdot f'(x)$		22. $\int \sin u\,du=-\cos u+c$	23. $\int \cos u\,du=\sin u+c$
5. $y=\sin f(x)\rightarrow {\frac {dy}{dx}}=[\cos f(x)]\cdot f'(x)$	6. $y=\cos f(x)\rightarrow {\frac {dy}{dx}}=[-\sin f(x)]\cdot f'(x)$		24. $\int \tan u\,du=-\ln |\cos u|+c$	25. $\int \cot u\,du=\ln |\sin u|+c$
7. $y=\tan f(x)\rightarrow {\frac {dy}{dx}}=[\sec ^{2}f(x)]\cdot f'(x)$	8. $y=\cot f(x)\rightarrow {\frac {dy}{dx}}=[-\csc ^{2}f(x)]\cdot f'(x)$		26. $\int \sec ^{2}u\,du=\tan u+c$	27. $\int \csc ^{2}u\,du=-\cot u+c$
9. $y=\sec f(x)\rightarrow {\frac {dy}{dx}}=[\sec f(x)\cdot \tan f(x)]\cdot f'(x)$	10. $y=\csc f(x)\rightarrow {\frac {dy}{dx}}=[-\csc f(x)\cdot \cot f(x)]\cdot f'(x)$		28. $\int \sec u\cdot \tan u\,du=\sec u+c$	29. $\int \csc u\cdot \cot u\,du=-\csc u+c$
11. $y=\ln f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{f(x)}}$	12. $y=\log _{a}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{f(x)\cdot \ln a}}$		30. $\int \sec u\,du=\ln |\sec u+\tan u|+c$	31. $\int \csc u\,du=-\ln |\csc u+\cot u|+c$
13. $y=e^{f(x)}\rightarrow {\frac {dy}{dx}}=e^{f(x)}\cdot f'(x)$	14. $y=\sin ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{\sqrt {1-[f(x)]^{2}}}}$		32. $\int e^{u}\,du=e^{u}+c$	33. $\int a^{u}\,du={\frac {a^{u}}{\ln a}}+c$
15. $y=\cos ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{\sqrt {1-[f(x)]^{2}}}}$	16. $y=\tan ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{1+[f(x)]^{2}}}$		34. $\int {\frac {du}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}{\frac {u}{a}}+c~({\mbox{if}}~a>0)$	35. $\int {\frac {du}{a^{2}-u^{2}}}={\frac {1}{a}}\tan ^{-1}{\frac {u}{a}}+c~({\mbox{if}}~a\neq 0)$
17. $y=\cot ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{1+[f(x)]^{2}}}$	18. $y=\sec ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{f(x)\cdot {\sqrt {[f(x)]^{2}-1}}}}$		36. $\int {\frac {du}{u{\sqrt {u^{2}-a^{2}}}}}={\frac {1}{a}}\sec ^{-1}{\frac {u}{a}}+c~({\mbox{if}}~a>0)$
19. $y=\csc ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{f(x)\cdot {\sqrt {[f(x)]^{2}-1}}}}$			37. Integration by Parts $\int u\,dv=uv-\int v\,du$