Ucale

# Fundamental Integration formulas

Since , $\displaystyle \frac { d }{ dx } \left\{ g\left( x \right) \right\} =f\left( x \right) \quad \Leftrightarrow \int { f\left( x \right) } dx=g\left( x \right) +c$

therefore, based upon this definition and various standard differentiation formulas, we obtain the following integration formulas:

 1. $\displaystyle \frac { d }{ dx } \left( \frac { { x }^{ n }+1 }{ n+1 } \right) ={ x }^{ n },\quad n\neq 1$ $\displaystyle \Rightarrow \int { { x }^{ n } } dx=\frac { { x }^{ n+1 } }{ n+1 } +C,\quad n\neq -1$ 2. $\displaystyle \frac { d }{ dx } \left( \log { x } \right)$ $\displaystyle \Rightarrow \int { \frac { 1 }{ x } dx } =\log { \left| x \right| } +C$ 3. $\displaystyle \frac { d }{ dx } \left( { e }^{ x } \right)$ $\displaystyle \Rightarrow \int { { e }^{ x } } dx={ e }^{ x }+C$ 4. $\displaystyle \frac { d }{ dx } \left( \frac { { a }^{ x } }{ \log _{ e }{ a } } \right) ={ a }^{ x },\quad a>0,\quad a\neq 1$ $\displaystyle \Rightarrow \int { { a }^{ x } } dx\quad =\frac { { a }^{ x } }{ \log _{ e }{ a } } +C$ 5. $\displaystyle \frac { d }{ dx } \left( -\cos { x } \right) =\sin { x }$ $\displaystyle \Rightarrow \int { \sin { x } } dx=-\cos { x } +C$ 6. $\displaystyle \frac { d }{ dx } \sin { x } =\cos { x }$ $\displaystyle \Rightarrow \int { \cos { x } } dx=\sin { x } +C$ 7. $\displaystyle \frac { d }{ dx } \tan { x } =\sec ^{ 2 }{ x }$ $\displaystyle \Rightarrow \int { \sec ^{ 2 }{ x } } dx=\tan { x } +C$ 8. $\displaystyle \frac { d }{ dx } -\cot { x } ={ cosec }^{ 2 }x$ $\Rightarrow \displaystyle \int { { cosec }^{ 2 }x } dx=-\cot { x } +C$ 9. $\displaystyle \frac { d }{ dx } \sec { x } =\sec { x } \tan { x }$ $\displaystyle \Rightarrow \int { \sec { x } \tan { x } } dx=\sec { x } +C$ 10. $\displaystyle \frac { d }{ dx } -cosecx=cosecx\quad \cot { x }$ $\displaystyle \Rightarrow \int { cosecx\quad \cot { x } } dx=-cosecx\quad +C$ 11. $\displaystyle \frac { d }{ dx } \log { \sin { x } } =\cot { x }$ $\displaystyle \Rightarrow \int { \cot { x } } dx=\log { \left| \sin { x } \right| } +C$ 12. $\displaystyle \frac { d }{ dx } -\log { \cos { x } } =\tan { x }$ $\displaystyle \Rightarrow \int { \tan { x } } dx=-\log { \left| \cos { x } \right| } +C$ 13. $\displaystyle \frac { d }{ dx } \log { \sec { x } \tan { x } } =\sec { x }$ $\displaystyle \Rightarrow \int { \sec { x } } dx=\log { \left| \sin { x } \tan { x } \right| } +C$ 14. $\displaystyle \frac { d }{ dx } \log { cosec\quad x-\cot { x } } =cosecx$ $\displaystyle \Rightarrow \int { cosecx } dx=\log { \left| \sec { x } \tan { x } \right| }$ 15. $\displaystyle \frac { d }{ dx } \left( \sin ^{ -1 }{ \frac { x }{ a } } \right) =\frac { 1 }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } }$ $\displaystyle \Rightarrow \int { \frac { 1 }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } dx=\sin ^{ -1 }{ \left( \frac { x }{ a } \right) } +C$ 16. $\displaystyle \frac { d }{ dx } \left( \cos ^{ -1 }{ \frac { x }{ a } } \right) =-\frac { 1 }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } }$ $\displaystyle \Rightarrow \int { -\frac { 1 }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } dx=\cos ^{ -1 }{ \left( \frac { x }{ a } \right) } +C$ 17. $\displaystyle \frac { d }{ dx } \left( \frac { 1 }{ a } \tan ^{ -1 }{ \frac { x }{ a } } \right) =\frac { 1 }{ { a }^{ 2 }+{ x }^{ 2 } }$ $\displaystyle \Rightarrow \int { \frac { 1 }{ { a }^{ 2 }+{ x }^{ 2 } } } dx=\frac { 1 }{ a } \tan ^{ -1 }{ \left( \frac { x }{ a } \right) } +C$ 18. $\displaystyle \frac { d }{ dx } \left( \frac { 1 }{ a } \cot ^{ -1 }{ \frac { x }{ a } } \right) =-\frac { 1 }{ { a }^{ 2 }+{ x }^{ 2 } }$ $\displaystyle \Rightarrow \int { -\frac { 1 }{ { a }^{ 2 }+{ x }^{ 2 } } } dx=\frac { 1 }{ a } \cot ^{ -1 }{ \left( \frac { x }{ a } \right) } +C$ 19. $\displaystyle \frac { d }{ dx } \left( \frac { 1 }{ a } \sec ^{ -1 }{ \frac { x }{ a } } \right) =\frac { 1 }{ x\sqrt { { x }^{ 2 }-{ a }^{ 2 } } }$ $\displaystyle \Rightarrow \int { \frac { 1 }{ x\sqrt { { x }^{ 2 }-{ a }^{ 2 } } } } dx=\frac { 1 }{ a } \sec ^{ -1 }{ \left( \frac { x }{ a } \right) } +C$ 20. $\displaystyle \frac { d }{ dx } \left( \frac { 1 }{ a } { cosec }^{ -1 }\frac { x }{ a } \right) =-\frac { 1 }{ x\sqrt { { x }^{ 2 }-{ a }^{ 2 } } }$ $\displaystyle \Rightarrow \int { -\frac { 1 }{ x\sqrt { { x }^{ 2 }-{ a }^{ 2 } } } } dx=\frac { 1 }{ a } { cosec }^{ -1 }\left( \frac { x }{ a } \right) +C$
April 18, 2019
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