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Find a function and constant such that a given integral equation holds

Find a function f and a constant c such that

    \[ \int_c^x t f(t) \, dt = \sin x - x \cos x - \frac{1}{2} x^2 \qquad \text{for all } x \in \mathbb{R}. \]


Let f(t) = \sin t - 1 and c = 0. Then,

    \begin{align*}  \int_c^x t f(t) \, dt &= \int_0^x (t \sin t - t) \, dt \\  &= \left. \left( \sin t - t \cos t - \frac{1}{2} t^2 \right) \right|_0^x \\  &= \sin x - x \cos x - \frac{1}{2} x^2. \end{align*}

2 comments

  1. Anonymous says:

    Since the question is to find a c, wouldn’t assuming the c to be pi/3 be incorrect? Wouldn’t it be more accurate if we take the integral on the LHS as P(x) from First Fundamental theorem, then differentiate it on both sides to find f(x), substitute this function as f(t) in the integration to find that in the definite integral, f(c) would be equal to the constant, thereby finding the value of c as pi/3? Sorry if I’m wrong.

  2. Anonymous says:

    I think it is easier to see the answer if you derivate both sides, so you’ll get x.f(x) = x(senx -1).
    Sorry if I am wrong about it, I’m still learning calculus

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