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{\bf Exam Question

Topic: DiffInt}

The function $f$ satisfies the equation
$$f(x)=\frac{\pi}{3}+\int_0^x \cos^2(f(t)\, dt.$$ By
differentiating the integral, find a differential equation for
$f(x).$ Hence find the function $f(x).$ \vspace{0.5in}

{\bf Solution}

$$f(x)=\frac{\pi}{3}+\int_0^x \cos^2(f(t)\, dt.$$
$$\frac{df}{dx}=\cos^2(f(x)),\ \ \mathrm{so}\ \
\sec^2(f)\frac{df}{dx}=1.$$ Therefore $\tan(f(x))=x+c.$

When $x=0,\ f(x)=\frac{\pi}{3},$ so $c=\sqrt3.$

Thus $f(x)=\tan^{-1}\left(x+\sqrt3\right).$


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