Mathematics

# Evaluate: $\displaystyle \int \dfrac{1}{1+tanx}dx$

##### SOLUTION
${l} \displaystyle \int{ \dfrac { { dx } }{ { 1+\tan x } } } \\$

$=\displaystyle \int{ \dfrac { { \cos{x}dx } }{ { \cos x+\sin x } } } \\$

$=\dfrac { 1 }{ 2 } \displaystyle \int { \dfrac { { \left( { \sin x+\cos x } \right) +\left( { \cos x-\sin x } \right) } }{ { \left( { \sin x+\cos x } \right) } } } dx \\$

$=\dfrac { x }{ 2 } +\dfrac { 1 }{ 2 } \displaystyle \int { \dfrac { { \left( { \cos x-\sin x } \right) } }{ { \sin x+\cos x } } } dx \\$

$\sin x+\cos x=t \\$

$=\dfrac { x }{ 2 } +\dfrac { 1 }{ 2 } \displaystyle \int { \dfrac { { dt } }{ t } } \\$

$I=\dfrac { x }{ 2 } +\dfrac { 1 }{ 2 } \log \left( { \sin x+\cos x } \right) +C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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