If $$fa+b+1$$−$$x=fx$$, for all x, where a and b are fixed positive real numbers, then $$1a+b$$∫$$abxfx+fx+1dxis$$ equal to

Question

If $$fa+b+1$$−$$x=fx$$, for all x, where a and b are fixed positive real numbers, then  $$1a+b$$∫$$abxfx+fx+1dxis$$ equal to

Infinity Learn · Accepted Answer

Given f  a  $$+$$  b  $$+$$  $$1$$  −  x  $$=$$  f  xit implies that  $$fx+1=fa+b$$−$$xI=1a+b$$∫$$abxfx+fx+1dx$$.                                      ......$$.1$$⇒$$I=1a+b$$∫$$aba+b$$−$$xfx+1+fxdx$$.                         ....$$.2Adding$$ (1) and (2)  $$2I=$$∫$$abfx+fx+1dx$$⇒$$2I=$$∫$$abfa+b$$−$$xdx+$$∫$$abfx+1dx$$⇒$$2I=2$$∫$$abfx+1dx$$⇒$$I=$$∫$$abfx+1dx$$                                           $$=$$∫$$a+1b+1fxdx$$

If $f (a + b + 1 - x) = f (x)$ , for all x, where $a$ and $b$ are fixed positive real numbers, then $\frac{1}{a + b} \int_{a}^{b} x (f (x)) + f (x + 1) d x$ is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If fa+b+1−x=fx, for all x, where a and b are fixed positive real numbers, then 1a+b∫abxfx+fx+1dxis equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If $f (a + b + 1 - x) = f (x)$ , for all x, where $a$ and $b$ are fixed positive real numbers, then $\frac{1}{a + b} \int_{a}^{b} x (f (x)) + f (x + 1) d x$ is equal to