Math Help - Definite Integration

1. Definite Integration

$F(x)$ is a differentiable function such that

$F'(a-x)=F'(x)$

for all $x$ satisfying $0\leq x\leq a$.

Evaluate

$\int_{0}^aF(x)dx$

and give an example of such a function.

2. Originally Posted by pankaj
$F(x)$ is a differentiable function such that

$F'(a-x)=F'(x)$

for all $x$ satisfying $0\leq x\leq a$.

Evaluate

$\int_{0}^aF(x)dx$

and give an example of such a function.
$f'(a-x)=f'(x)$ gives us $f(x)+f(a-x)=c=\text{constant}.$ put $x=a/2$ to get $c=2f(a/2).$ now substitute $x \to a-x$ to get $\int_0^af(x) \ dx = \int_0^a f(a-x) \ dx = \int_0^a (c-f(x)) \ dx.$

thus: $\int_0^a f(x) \ dx = \frac{ac}{2}=af(a/2).$ an example is $f(x)=x.$ here we have $c=a$ and $\int_0^a x \ dx = \frac{a^2}{2}=a \cdot a/2=af(a/2).$