12th Harvard MIT Calculus Problem 2


The differentiable function F:\mathbb{R}\to \mathbb{R} satisfied F(0)=-1 and \frac{d}{dx}F(x)=\sin (\sin (\sin (\sin (x))))\cos (\sin (\sin (x)))\cos (\sin (x))\cos (x)

Find F(x) as a function of x.

Solution

Substitution u=\sin (\sin (\sin (x)))

then du={{(\sin (\sin (x)))}^{'}}\cos (\sin (\sin (x)))dx=\cos (\sin (\sin (x)))\cos (\sin x)\cos xdx

So we get F(x)=\int{\sin udu=-\cos u+c}

since we know that F(0)=-1 then we get c=0

Therefore F(x)=\cos (\sin (\sin (\sin (x))))

😀 Thank you Pablo for comment about F(x)=\cos x 😀

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About Bunchhieng

I like math, programming and web design!!
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