Mathematics

The integral  $$\int \cos \left( \log _ { e } x \right) d x$$  is equal to :
(where  $$C$$  is a constant of integration)


ANSWER

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


SOLUTION
$${ I }=\int  \cos  (\ell { n }{ x }){ d }{ x }$$

$${ I }=\cos  (\ln {  } { x })\cdot { x }+\int  \sin  (\ell { n }{ x }){ d }{ x }$$

$$\cos ( \ell n x ) x + \left[ \sin ( \ell n x ) \cdot x - \int \cos ( \ell n x ) d x \right]$$

$${ I }=\dfrac { { x } }{ 2 } [\sin  (\ell { n }{ x })+\cos  (\ell { n }{ x })]+{ C }$$
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Single Correct Medium Published on 17th 09, 2020
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