Mathematics

# $\int { \cos { \left( \log { x } \right) } } dx=............\quad +\quad c$

$\dfrac { x }{ 2 } \left[ \cos { \left( \log { x } \right) } +\sin { \left( \log { x } \right) } \right]$

##### SOLUTION

Consider the given integral.

$I=\int{\cos \left( \log x \right)dx}$

Let $t=\log x$

$\dfrac{dt}{dx}=\dfrac{1}{x}$

$xdt=dx$

Therefore,

$I=\int{{{e}^{t}}\cos tdt}$

$I=\cos t{{e}^{t}}-\int{\left( -\sin t \right){{e}^{t}}}dt$

$I={{e}^{t}}\cos t+\int{\left( \sin t \right){{e}^{t}}}dt$

$I={{e}^{t}}\cos t+\sin t{{e}^{t}}-\int{\cos t{{e}^{t}}}dt$

$I={{e}^{t}}\cos t+\sin t{{e}^{t}}-I$

$2I={{e}^{t}}\left( \cos t+\sin t \right)+C$

$I=\dfrac{{{e}^{t}}}{2}\left( \cos t+\sin t \right)+C$

On putting the value of $'t'$, we get

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

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

Hence, this is the answer.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
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