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Integrate : $\int \cos (\log x)dx$

$(a)\;\frac{1}{2} (\cos (\log x)+\sin (\log x))+c \qquad(b)\;\frac{1}{2} (x \cos (\log x)+ \sin (\log x))+c $$(c)\;\frac{1}{2} (x \cos (\log x)+x \sin (\log x))+c \qquad (d)\;\frac{1}{2} (\cos (\log x)+x \sin (\log x))+c $

1 Answer

$\log x =t$
differentiate with respect to x
$\large\frac{1}{x} $$dx=dt$
$dx= x.dt$
$ dx= e^t.dt$
$I= \int e^t.\cot t dt$
$I= \int e^t. \cos t + \int e^t. \sin t dt$
$I= e^t. \cot t +e^t. \sin t -\int e^t \cos t dt$
$I= \large\frac{1}{2} $$[e^t. \cot t +e^t. \sin t]+c$
$\large\frac{1}{2}$$ (x \cos (\log x)+x \sin (\log x))+c$
Hence c is the correct answer.
answered Dec 24, 2013 by meena.p