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Question 13.1: Given  dy / dx = cos x − x, find y....

Given  \frac{\mathrm{d} y}{\mathrm{~d} x}=\cos x-x, \text { find } y .

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We need to find a function which, when differentiated, yields \cos x  −  x. Differentiating sin x yields \cos x, while differentiating −x²/2 yields −x. Hence,

y=\int(\cos x-x) \mathrm{d} x=\sin x-\frac{x^2}{2}+c

where c is the constant of integration. Usually brackets are not used and the integral is written simply as  \int \cos x-x d x.

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