Question
Mathematics Question on Sequence and series
What is ∫0af(a−x)f(x)+f(a−x)dx equal to?
A
(A) a
B
(B) 2a
C
(C) 0
D
(D) a2
Answer
(D) a2
Explanation
Solution
Explanation:
As we know that ∫abf(x)dx=∫abf(a+b−x)dx I=∫0af(a−x)f(x)+f(a−x)dx…… (1)So, we have lower limit a=0, upper limit b=a⇒I=∫0af[(0+a)−(a−x)]f[(a+0)−x]+f[(a+0)−(a−x)]dx⇒I=∫0af(x)f(a−x)+f(x)dx……… (2)Now adding equation (1) and equation (2),⇒2I=∫0af(a−x)f(x)+f(a−x)dx+∫0af(x)f(a−x)+f(x)dx⇒2I=∫0a1dx⇒2I=a∴I=a2Hence, the correct option is (D).