Equations on Integration!

Calculus Level 5

0xtf(xt) dt=0xf(t) dt+sin(x)+cos(x)x1\large{\int_0^x tf(x-t)\ \mathrm{d}t = \int_0^x f(t)\ \mathrm{d}t + \sin(x) + \cos(x) -x-1}

Find a continuous function f(x)f(x) such that for all real xx, the above equation is satisfied. Determine the value of f(π6)f \left( \frac{\pi}6 \right) to three decimal places.

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