Some Special Integrals

Some of the special integrals in derived form to be used where it will be applicable :

1. $\displaystyle \int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } }$	$\displaystyle =\frac { 1 }{ a } \tan ^{ -1 }{ \left( \frac { x }{ a } \right) } +c$
2. $\displaystyle \int { \frac { dx }{ { x }^{ 2 }-{ a }^{ 2 } } }$	$\displaystyle =\frac { 1 }{ 2a } \log { \left\| \frac { x-a }{ x+a } \right\| } +c$
3. $\displaystyle \int { \frac { dx }{ { a }^{ 2 }-{ x }^{ 2 } } }$	$\displaystyle =\frac { 1 }{ 2a } \log { \left\| \frac { a+x }{ a-x } \right\| } +c$
4. $\displaystyle \int { \frac { dx }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } }$	$\displaystyle =\sin ^{ -1 }{ \left( \frac { x }{ a } \right) } +c$
5. $\displaystyle \int { \frac { dx }{ \sqrt { { x }^{ 2 }-{ a }^{ 2 } } } }$	$\displaystyle =\log { \left\| x+\sqrt { { x }^{ 2 }-{ a }^{ 2 } } \right\| } +c$
6. $\displaystyle \int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx$	$\displaystyle =\log { \left\| x+\sqrt { { x }^{ 2 }-{ a }^{ 2 } } \right\| } +c$
7. $\displaystyle \int { \sqrt { { a }^{ 2 }+{ x }^{ 2 } } dx }$	$\displaystyle =\frac { 1 }{ 2 } x\sqrt { { a }^{ 2 }+{ x }^{ 2 } } +\frac { 1 }{ 2 } { a }^{ 2 }\sin ^{ -1 }{ \left( \frac { x }{ a } \right) } +c$
8. $\displaystyle \int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } dx }$	$\displaystyle =\frac { 1 }{ 2 } x\sqrt { { a }^{ 2 }+{ x }^{ 2 } } +\frac { 1 }{ 2 } { a }^{ 2 }\log { \left\| x+\sqrt { { a }^{ 2 }+{ x }^{ 2 } } \right\| } +c$
9. $\displaystyle \int { \sqrt { { x }^{ 2 }-{ a }^{ 2 } } dx }$	$\displaystyle =\frac { 1 }{ 2 } x\sqrt { { a }^{ 2 }+{ x }^{ 2 } } -\frac { 1 }{ 2 } { a }^{ 2 }\log { \left\| x+\sqrt { { x }^{ 2 }-{ a }^{ 2 } } \right\| } +c$

Some Important Substitution

$\displaystyle { a }^{ 2 }+{ x }^{ 2 }$	$\displaystyle x=a\tan { \theta } \quad or\quad a\cot { \theta }$
$\displaystyle { a }^{ 2 }-{ x }^{ 2 }$	$\displaystyle x=a\sin { \theta } \quad or\quad a\cos { \theta }$
$\displaystyle { x }^{ 2 }-{ a }^{ 2 }$	$\displaystyle x=a\sec { \theta } \quad or\quad acosec\theta$
$\displaystyle \sqrt { \frac { a-x }{ a+x } } \quad or\quad \sqrt { \frac { a+x }{ a-x } }$	$\displaystyle a\cos { 2\theta }$
$\displaystyle \sqrt { \frac { x-\alpha }{ \beta -x } } \quad or\quad \sqrt { \left( x-\alpha \right) \left( x-\beta \right) }$	$\displaystyle x=\alpha \cos ^{ 2 }{ \theta } +\beta \sin ^{ 2 }{ \theta } +c$