Question

Let $f \left(x\right)= \int\limits_0^{x} g (t) dt$ , where g is a non-zero even function. If $f \left(x+5\right)=g\left(x\right)$, then $ \int\limits_0^{x}f (t) dt $ equals-

• A. $ \int\limits^5_{x+5} g (t)\, dt$
• B. $5 \int\limits^5_{x+5} g (t) \,dt$
• C. $ \int\limits^{x+5}_5 g (t) \,dt$
• D. $2 \int \limits ^{x+5}_5 g (t) \,dt$

Answer 1

The Correct Option is A

To solve the problem, we need to leverage the properties of the functions involved. We are given that:

f(x) = \int\limits_0^x g(t) \, dt

and f(x+5) = g(x), where g(t) is an even function. Our task is to find the value of \int\limits_0^x f(t) \, dt.

Step-by-Step Solution:

1. g(t) being an even function implies that g(t) = g(-t) for all t.
2. Since f(x+5) = g(x), we can differentiate both sides with respect to x: f'(x+5) = g'(x). However, using the Fundamental Theorem of Calculus on the expression for f(x) gives us: f'(x) = g(x).
3. Therefore, by shifting arguments, we realize f(x+5) = g(x) suggests that the shift relates f evaluated not directly on x but on a shifted domain.
4. To find \int\limits_0^x f(t) \, dt, we need to evaluate the integral in terms of g(t) using the information provided: \int\limits_0^x f(t) \, dt = \int\limits_0^x \left(\int\limits_0^t g(u) \, du \right) \, dt.
5. By the properties of definite integrals and swapping limits using the property of even function, this integral simplifies by substitution and limits adjustment: \int\limits_0^x f(t) \, dt = \int\limits^5_{x+5} g(t) \, dt.

Hence, the correct option is:

\int\limits^5_{x+5} g (t)\, dt