Mathematics

# Evaluate: $\int {\dfrac{{dx}}{{\left( {5 - 8x - {x^2}} \right)}}dx}$

##### SOLUTION
$\begin{array}{l} We\, have \\ \int { \dfrac { { dx } }{ { \left( { 5-8x-{ x^{ 2 } } } \right) } } dx } =-\int { \dfrac { { dx } }{ { \left( { { x^{ 2 } }+8x-5 } \right) } } } \\ =-\int { \dfrac { { dx } }{ { \left\{ { \left( { { x^{ 2 } }+8x+16 } \right) -21 } \right\} } } } \\ =-\int { \dfrac { { dx } }{ { \left\{ { { { \left( { x+4 } \right) }^{ 2 } }-{ { \left( { \sqrt { 21 } } \right) }^{ 2 } } } \right\} } } } \\ =\int { \dfrac { { dx } }{ { \left\{ { { { \left( { \sqrt { 21 } } \right) }^{ 2 } }-{ { \left( { x+4 } \right) }^{ 2 } } } \right\} } } } \\ =\int { \dfrac { { dt } }{ { \left\{ { { { \left( { \sqrt { 21 } } \right) }^{ 2 } }-{ t^{ 2 } } } \right\} } } \, \, \, \, \, where\, \, \left( { x+4 } \right) =t } \\ =\dfrac { 1 }{ { 2\sqrt { 21 } } } .\log \left| { \dfrac { { \sqrt { 21 } +t } }{ { \sqrt { 21 } -t } } } \right| +C \\ =\dfrac { 1 }{ { 2\sqrt { 21 } } } .\log \left| { \dfrac { { \sqrt { 21 } +4+x } }{ { \sqrt { 21 } -4-x } } } \right| +C. \end{array}$

Hence, this is the required answer.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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