Mathematics

# Evaluate:$\displaystyle \int \dfrac{1}{5-4 \cos x}.dx$

##### SOLUTION
$\quad \int { \dfrac { 1 }{ 5-4\cos x } dx\quad } \\ =\int { \dfrac { dx }{ 5\left( { \cos }^{ 2 }\dfrac { x }{ 2 } +{ \sin }^{ 2 }\dfrac { x }{ 2 } \right) +4\left( { \cos }^{ 2 }\dfrac { x }{ 2 } -{ \sin }^{ 2 }\dfrac { x }{ 2 } \right) } } \\ =\int { \dfrac { dx }{ 5{ \cos }^{ 2 }\dfrac { x }{ 2 } +5{ \sin }^{ 2 }\dfrac { x }{ 2 } +4{ \cos }^{ 2 }\dfrac { x }{ 2 } -4{ \sin }^{ 2 }\dfrac { x }{ 2 } } } \\ =\int { \dfrac { dx }{ 9{ \cos }^{ 2 }\dfrac { x }{ 2 } +{ \sin }^{ 2 }\dfrac { x }{ 2 } } } \\ =\int { \dfrac { { \sec }^{ 2 }\dfrac { x }{ 2 } }{ 9+{ \tan }^{ 2 }\dfrac { x }{ 2 } } dx\quad } \\ Now,\quad let\\ \tan\dfrac { x }{ 2 } =t\\ differentiating\quad w.r.t\quad x,\quad we\quad get\\ \quad \quad { \sec }^{ 2 }\dfrac { x }{ 2 } \times \dfrac { 1 }{ 2 } dx=dt\\ \Rightarrow { \sec }^{ 2 }\dfrac { x }{ 2 } dx=2dt\\ now\quad putting\quad these\quad values\quad in\quad the\quad above\quad equation\quad we\quad get,\\ \quad \quad \int { \dfrac { { \sec }^{ 2 }\dfrac { x }{ 2 } }{ 9+{ \tan }^{ 2 }\dfrac { x }{ 2 } } dx\quad } \\ =2\int { \dfrac { dt }{ 9+{ t }^{ 2 } } } \\ =2\int { \dfrac { dt }{ { \left( 3 \right) }^{ 2 }+{ \left( t \right) }^{ 2 } } } \\ =\dfrac { 2 }{ 3 } { \tan }^{ -1 }\left( \dfrac { t }{ 3 } \right) \\ =\dfrac { 2 }{ 3 } { \tan }^{ -1 }\left( \dfrac { \tan\dfrac { x }{ 2 } }{ 3 } \right) +C$

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

#### Realted Questions

Q1 Subjective Hard
Evaluate $\int {\left( {\log \left( {\log x} \right) + \frac{1}{{{{(\log x)}^2}}}} \right)dx}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int(\log x)^{2}dx=$
• A. $x[(\log x)^{2}+2$ $\log x$ $+2]+c$
• B. $[(\log x)^{2}-2$ $\log x$ $+2]+c$
• C. $[(\log x)^{2}+2$ $\log x$ $+2]+c$
• D. $x[(\log x)^{2}-2$ $\log x$ $+2]+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Solve$\displaystyle \int_{0}^{\pi /2}\left ( 2\log \sin x-\log \sin 2x \right )dx$
• A. $\dfrac {\pi}{2}\log 2$
• B. $\dfrac {\pi}{4}\log 2$
• C. None
• D. $-\dfrac {\pi}{2}\log 2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve $\displaystyle\int { \dfrac { x }{ \sqrt { x+4 } } dx }$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$I=\displaystyle \int \dfrac{(x+a)^3}{x^3}dx$ is equal to:
• A. $x^2+ 3a \log x -\dfrac{3a^2}{x} - \dfrac{a^3}{2x^2}+c$
• B. $x^3+ 3a \log x -\dfrac{2a^2}{x} - \dfrac{3a^3}{2x^2}+c$
• C. $1+ 2a \log x -\dfrac{2a^2}{x} - \dfrac{3a^2}{2x^2}+c$
• D. $x+ 3a \log x -\dfrac{3a^2}{x} - \dfrac{a^3}{2x^2}+c$