Derivatives and Integrals

Calculus Level pending

Let f ( x ) = 2 ( cos ⁡ x − sin ⁡ x ) + ( 2 − x ) ( cos ⁡ 2 x sin ⁡ x ) f(x) = 2(\cos x - \sin x) + (2-x) \left( \dfrac{\cos^2 x}{\sin x} \right) . Find d d x ∫ f ( x ) d x \displaystyle \dfrac d{dx} \int f(x) \, dx .

There are infinitely many solutions ( − sin ⁡ x − cos ⁡ x ) − cos ⁡ 2 x sin ⁡ x − ( 2 − x ) cos ⁡ 3 x sin ⁡ 2 x − 2 ( 2 − x ) cos ⁡ x (-\sin x - \cos x) - \cos^2 x \sin x - (2-x) \cos^3 x \sin^2 x - 2(2-x) \cos x 2 ( cos ⁡ x − sin ⁡ x ) + ( 2 − x ) ( cos ⁡ 2 x sin ⁡ x ) 2(\cos x - \sin x) + (2-x) \left( \dfrac{\cos^2 x}{\sin x} \right) 2 ( − cos ⁡ x − cos ⁡ x ) + ( − 1 ) ( − d f r a c 2 sin ⁡ x cos ⁡ x ) 2(-\cos x - \cos x) + (-1) \left( -dfrac{2\sin x}{\cos x} \right)

Zachary Wolf by Zachary Wolf , 20, USA

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1 solution

J D
Jul 20, 2016

By FTC the derivative of the integral is the original function

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Its fairly obvious and if you pay attention as you read it is clear that if you take the derivative of an integral you will return to your original equation. The equation is completely arbitrary and you can place literally anything in place of it.

Zachary Wolf - 4 years, 11 months ago

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