Recovering a Function from Integral Conditions

The procedure:

• Write the unknown function with its parameters, e.g. $f(x)=Pe^x+Qe^{2x}+Re^{3x}$.
• Convert each condition into an equation: a value $f(0)$, an integral $\int_0^c f$, or a derivative $f'(0)=P+2Q+3R$.
• Solve the resulting linear system for the parameters, then answer the specific question asked.

When the unknown is the FUNCTION, not its coefficients — the reciprocal-argument trick: a relation such as $3f(x)+f\!\big(\tfrac1x\big)=g(x)$ holds for every $x$, so replace $x$ by $\tfrac1x$ to get a second relation $3f\!\big(\tfrac1x\big)+f(x)=g\!\big(\tfrac1x\big)$; solve the pair as simultaneous equations in the two unknowns $f(x)$ and $f(1/x)$ (here $8f(x)=3g(x)-g(1/x)$). The same pairing works for $x\to1-x$, $x\to-x$ and $x\to a-x$ — whichever substitution maps the given arguments onto each other. Then integrate the recovered $f$.