Question

Let T > 0 be a fixed real number. Suppose, f is a continuous function such that for all $x \in R. f(x+T)=f(x). \, If \, I = \int_0^T f(x)dx$ then the value of $\int_3^{3+3T} f(2x)dx$

• A. $\frac{3}{2}I$
• B. I
• C. 3I
• D. 6I

Answer 1

The Correct Option is C

The correct answer is :3I
Given that;
\(I=\int^T_0f(x)dx\), \(f\) is continuous function such that \(x\in{R}\),\(f(x+T)=f(x)\), \(8(T>0)\)
To find;\(\int^{3+3T}_{3}f(2x)dx\)
take \(2x=t\space \therefore dx=\frac{dt}{2}\)
\(\therefore \frac{1}{2}\int^{b(1+T)}_{b}f(t)dt=\frac{6}{2}\int^T_0f(t)dt\)
\(=3\int^T_0f(x)dt\)
\(=3I\)