Mathematics

# Solve : $\int \cos (log x)dx$ ?

##### SOLUTION
Given, $\int { \cos { (\log { x } ) } } dx$
Let, $\log { x } =t\\ \Rightarrow x={ e }^{ t }$
and $\dfrac { 1 }{ x } dx=dt\\ \Rightarrow dx=xdt\\ \Rightarrow dx={ e }^{ t }dt$
So, we can write the integral as
$\int { \cos { (\log { x } ) } } dx\\ =\int { \cos { (t) } { e }^{ t } } dt\\ =\dfrac { 1 }{ 2 } \int { { e }^{ t } } (2\cos { t } )dt\\ =\dfrac { 1 }{ 2 } \int { { e }^{ t } } \left[ (\sin { t } +\cos { t } )+(\cos { t } -\sin { t } ) \right] dt$
We know that, $\int { { e }^{ t } } \left[ f\left( t \right) +f^{ ' }\left( t \right) \right] dt={ e }^{ t }\left[ f\left( t \right) \right] +C$
Thus, $f\left( t \right) =\sin { t } +\cos { t }$ and $f^{ ' }\left( t \right) =\cos { t } -\sin { t }$
$\therefore \dfrac { 1 }{ 2 } \int { { e }^{ t } } \left[ (\sin { t } +\cos { t } )+(\cos { t } -\sin { t } ) \right] dt\\ =(\dfrac { 1 }{ 2 } ){ e }^{ t }\left[ \sin { t } +\cos { t } \right] +C\\ =\dfrac { 1 }{ 2 } x\left[ \sin { (\log { x } ) } +\cos { (\log { x } ) } \right] +C$
Hence, $\int { \cos { (\log { x } ) } } dx=\dfrac { 1 }{ 2 } x\left[ \sin { (\log { x } ) } +\cos { (\log { x } ) } \right] +C.$

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

#### Realted Questions

Q1 One Word Hard
State whether True=1 or False=0
$\displaystyle { \int { \frac { x^{ 2 } }{ (x^{ 2 }+1)(x^{ 2 }+4) } dx } }=\frac { -1 }{ 3 } \tan ^{ -1 }{ x } +\frac {2 }{ 3 } \tan ^{ -1 }{ \left(\frac { x }{ 2 }\right) } +C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\int { \cfrac { 1 }{ x\sqrt { 1-{ x }^{ 3 } } } } dx\quad$ is equal to

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate: $\displaystyle \int_{0}^{\pi/4}\sec^{6}x \ dx$
• A. $\dfrac {8}{15}$
• B. $\dfrac {35}{8}$
• C. $\dfrac {44}{15}$
• D. $\dfrac {28}{15}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\int _{ 0 }^{ 1 }{ \frac { x }{ 1+\sqrt { x } } dx= }$
• A. $\frac {5}{3}+\log {4}$
• B. $\frac {5}{3}\log {4}$
• C. $\frac {3}{5}-\log {4}$
• D. $\frac {5}{3}-\log {4}$

The value of $\int _{ 0 }^{ { \pi }/{ 2 } }{ \cfrac { dx }{ 1+\tan ^{ 3 }{ x } } }$ is