WBJEE · Maths · Definite Integration
Which of the following statements are true?
- A If \(f(x)\) be continuous and periodic with periodicity \(T\), then \(I=\int_a^{a+T} f(x) d x\) depend on ' \(a\) '.
- B If \(f(x)\) be continuous and periodic with periodicity \(T\), then \(I=\int_a^{a+T} f(x) d x\) does not depend on ' \(a\) '.
- C Let \(f(x)=\left\{\begin{array}{l}1 \text {, if } x \text { is rational } \\ 0, \text { if } x \text { is irrational }\end{array}\right.\), then \(f\) is periodic of the periodicity \(T\) only if \(T\) is rational
- D \(f\) defined in (C) is periodic for all \(T\)
Answer & Solution
Correct Answer
(D) \(f\) defined in (C) is periodic for all \(T\)
Step-by-step Solution
Detailed explanation
Hint: (A) \(I=\int_a^{a+T} f(x) d x=\int_0^T f(x) d x\) Independent of a (B) \(I=\int_a^{a+T} f(x) d x=\int_0^T f(x) d x\) does not depend on ' \(a\) ' (C) \(f(x)=\left\{\begin{array}{l}1, x \in Q \\ 0, x \notin Q\end{array}\right.\) is periodic for all \(T\) with undefined…
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