TITLE: Show that the value of a definite integral is unity
QUESTION [6 upvotes]: $$ \int_2^4\frac{\sqrt{\log(9-x)}}{\sqrt{\log(9-x)}+\sqrt{\log(3+x)}}dx=1$$

REPLY [3 votes]: $ \int^a_b f(x)\,dx $ = $ \int^a_b f(a+b-x)\,dx $
 We can prove this by changing the dummy variable x to a+b-x, we get the integrand as $-f(a+b-x)\,dx$ and by changing the limits accordingly with $a+b-b$ and $a+b-a$, i.e. Now the integral becomes $-\int^b_af(a+b-x)dx$=$ \int^a_b f(a+b-x)dx $
The function given in the question satisfies the property that $f(x)+f(6-x)=1$. Let the integral be $I$ then $2I= \int^a_b f(x)\,dx+ \int^a_b f(a+b-x)dx=\int^a_b f(x)+f(a+b-x)\,dx=\int^a_b\,dx=a-b$, In your case $a-b=2$, hence $2I=2$. So $I=1$.