integral question help me please

Question

$$\int_0^1 \frac{f(x)} {f(x)+f(1-x)}dx$$ Thank you very much

Same trick as here: http://math.stackexchange.com/questions/82489/how-can-i-calculate-int-0-pi-2-frac-sin3-t-sin3-t-cos3-tdt — Hans Lundmark, Jul 10 '15 at 07:24

score 3 · Answer 1 · answered Jul 10 '15 at 07:26

3

Let's try substitution $y=1-x$: $$ I=\int_0^1 \frac{f(x)}{f(x)+f(1-x)}dx=\int_0^1 \frac{f(1-y)}{f(1-y)+f(y)}dy; $$ hence, $$2I=\int_0^1 \frac{f(x)+f(1-x)}{f(x)+f(1-x)}dx=1\Longrightarrow I=\frac12$$

answered Jul 10 '15 at 07:26

Michael Galuza

4,730

score 2 · Accepted Answer · answered Jul 10 '15 at 07:26

use $x\to 1-x$: \begin{align} I&=\int_0^1 \frac{f(x)} {f(x)+f(1-x)}dx\\ &=\int_0^1 \frac{f(1-x)} {f(x)+f(1-x)}dx \end{align} hence \begin{align} 2I=I+I&=\int_0^1 \frac{f(x)} {f(x)+f(1-x)}dx+\int_0^1 \frac{f(1-x)} {f(x)+f(1-x)}dx\\ &=\int_0^1 dx=1 \end{align} therefore $I=\frac12$.

integral question help me please

2 Answers2