Finding the Function at A Point from the Integral

Problem 188: Find $f(4)$ if

(1) $\begin{equation*} \int_{0}^{x} f(t) dt = x \cos \pi x \end{equation*}$

Solution: Using the Fundamental Theorem of Calculus (Part 1), we have

(2) $\begin{align*} \int_{0}^{x} f(t) dt & = x \cos \pi x \\ \frac{d}{dx} \left( \int_{0}^{x} f(t) dt \right) & = \frac{d}{dx} \left( x \cos \pi x\right) \\ f(x) & = \cos \pi x - \pi x \sin \pi x \\ f(4) & = \cos(4 \pi) - 4\pi \sin(4\pi).\end{align*}$

Hence,

(3) $\begin{equation*} f(4) = \cos(4 \pi) - 4\pi \sin(4 \pi).\end{equation*}$

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Finding the Function at A Point from the Integral

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