IIT-JEE 1984Question 1
Given a function $f(x)$ such that (i) it is integrable over every interval on the real line and (ii) $f(t+x)=f(x)$, for every $x$ and a real $t$, then show that the integral $\displaystyle\int_a^{a+t}f(x)\,dx$ is independent of $a$.
Definite Integration
Answerproved
Solution
Concept. Split $\int_a^{a+t}$ at $0$ and $t$; shifting the last piece by $t$ cancels the first piece. (If $f$ is also continuous, a shorter route is $F(a)=\int_a^{a+t}f$, $F'(a)=f(a+t)-f(a)=0$.) Formulas used. (1) $\int_a^{a+t}=\int_a^0+\int_0^t+\int_t^{a+t}$(2) $x=y+t$: $\int_t^{a+t}f(x)\,dx=\int_0^af(y+t)\,dy=\int_0^af(y)\,dy$ (periodicity)Step 1 - Split. By (1), $$\int_a^{a+t}f=\int_a^0f+\int_0^tf+\int_t^{a+t}f.$$ Step 2 - Shift. By (2), the last piece is $\int_0^af$, which cancels $\int_a^0f$. Step 3 - Conclude. $\int_a^{a+t}f(x)\,dx=\int_0^tf(x)\,dx$, which does not involve $a$. Hence proved. Answer: proved