The Green Book of Math Problems Question 5

Question

Evaluate the following integral where it is noted that the denominator never vanishes over the interval. No assumption is made on the continuity of f(x) or its derivatives.

$$\int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} dx$$

Answer 1

I evaluated it and the answer is

$$\frac{a}{2}$$

Answer 2

$$\int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} dx$$

Consider that the interval of integration is divided up into n equal segments with n+1 end points $x_0,x_1,..x_n.$ The y values are given by $y_0=f(x_0),y_1=f(x_1)...y_n=f(x_n)=f(a).$

$$\int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} dx$$ can be approximated by the sum $$\frac{a}{n} \left( \frac{y_0}{y_0+y_n}+\frac{y_1}{y_1+y_{n-1}}+ \ldots \frac{y_n}{y_0+y_n} \right)$$ Thus there are n/2 fractions that have the same denominator which can be combined to obtain $$\frac{a}{n} \left( \frac{y_0+y_n}{y_0+y_n}+\frac{y_1+y_{n-1}}{y_1+y_{n-1}} + \ldots \right) =\frac{a}{n} \frac{n}{2}=\frac{a}{2} $$ Notice that the n disappears from the summation. Even if the number of terms was infinite the answer is still the same. This means that not only is the integral equal to this value but also the discrete sum.