Mathematics

$$\int $$ sin x d(cos x) is equal to


ANSWER

$$-\dfrac{1}{4}cos2x+c$$


SOLUTION

Consider the given equation:

$$ :\int{\sin x\,d}(\cos x)dx $$

$$ =\int{\sin (x)\cos (x)dx} $$

$$ =\int{\dfrac{\sin (2x)}{2}}dx $$

$$ =\dfrac{1}{2}\int{\sin (2x)dx} $$

$$ \,\,\,\text{put}\,\text{then}\,\,v=2x $$

$$ \,\,\,\text{so}\,\text{that}\,\,dv=2dx $$

$$ =\dfrac{1}{4}\int{\sin (v).(2)dx} $$

$$ =\dfrac{1}{4}\int{\sin (v)dx} $$

$$ =-\dfrac{1}{4}\cos (v)+c $$

$$ =-\dfrac{1}{4}\cos (2x)+c $$

$$ Hence\,,this\,\,is\,\,the\,\,answer. $$

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Single Correct Medium Published on 17th 09, 2020
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