Ucale

# Some Special Integrals

### 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

 $\displaystyle { a }^{ 2 }+{ x }^{ 2 }$ $\displaystyle x=a\tan { \theta } \quad or\quad a\cot { \theta }$ $\displaystyle { a }^{ 2 }-{ x }^{ 2 }$ $\displaystyle x=a\sin { \theta } \quad or\quad a\cos { \theta }$ $\displaystyle { x }^{ 2 }-{ a }^{ 2 }$ $\displaystyle x=a\sec { \theta } \quad or\quad acosec\theta$ $\displaystyle \sqrt { \frac { a-x }{ a+x } } \quad or\quad \sqrt { \frac { a+x }{ a-x } }$ $\displaystyle a\cos { 2\theta }$ $\displaystyle \sqrt { \frac { x-\alpha }{ \beta -x } } \quad or\quad \sqrt { \left( x-\alpha \right) \left( x-\beta \right) }$ $\displaystyle x=\alpha \cos ^{ 2 }{ \theta } +\beta \sin ^{ 2 }{ \theta } +c$
April 18, 2019
Which class you are presently in?
Choose an option. You can change your option at any time.
You will be solving questions and growing your critical thinking skills.