# Definite Integration

• Jun 11th 2009, 06:17 PM
pankaj
Definite Integration
$\displaystyle F(x)$ is a differentiable function such that

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

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

Evaluate

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

and give an example of such a function.
• Jun 11th 2009, 07:21 PM
NonCommAlg
Quote:

Originally Posted by pankaj
$\displaystyle F(x)$ is a differentiable function such that

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

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

Evaluate

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

and give an example of such a function.

$\displaystyle f'(a-x)=f'(x)$ gives us $\displaystyle f(x)+f(a-x)=c=\text{constant}.$ put $\displaystyle x=a/2$ to get $\displaystyle c=2f(a/2).$ now substitute $\displaystyle x \to a-x$ to get $\displaystyle \int_0^af(x) \ dx = \int_0^a f(a-x) \ dx = \int_0^a (c-f(x)) \ dx.$

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