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Fundamental Integration formulas

Since , $latex \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 &s=1 $

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

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