If f(x) + f(-x) = 0, then $\int_a^x {f(t)dt} $ is

Question

Prepp Team · Accepted Answer

Understanding Odd Functions in Calculus and Their Properties

In calculus, an odd function $f(x)$ is characterized by the property $f(-x) = -f(x)$ for all $x$ in its domain. This property reflects a specific type of symmetry. Examples of odd functions include $x$, $x^3$, and $\sin(x)$.

The question asks about the nature of the function obtained by performing a definite integral of an odd function $f(t)$ from a fixed lower limit $a$ to a variable upper limit $x$. Let's call this resulting function $G(x)$. The property of this resulting function is what we need to determine.

We have $G(x) = \int_a^x f(t)dt$.

Determining the Function Type: Odd or Even

To classify $G(x)$ as an odd function or an even function, we examine its symmetry property. A function $G(x)$ is an odd function if $G(-x) = -G(x)$ for all $x$. Conversely, it's an even function if $G(-x) = G(x)$.

Analyzing the Definite Integral of an Odd Function $\int_a^x f(t)dt$

Let's consider the expression for $G(-x)$:

$G(-x) = \int_a^{-x} f(t)dt$

The integration process and the limits determine the specific form of the function $G(x)$. The property of the integrand being an odd function significantly influences the symmetry of the resulting integral function. When evaluating $G(-x)$ and comparing it to $G(x)$, we utilize the definition of an odd function $f(t)$ and the fundamental properties of integration.

Resulting Function Property for Odd Integrands

For an odd function $f(x)$, the definite integral $\int_a^x f(t)dt$ defines a function $G(x)$ such that evaluating $G(-x)$ yields a specific relationship with $G(x)$. Based on the property of integrating odd functions and the structure of the integral $\int_a^x$, the resulting function $G(x)$ is found to satisfy the condition $G(-x) = -G(x)$.

This condition, $G(-x) = -G(x)$, is the defining property of an odd function.

Conclusion: The Odd Function Integral Result

Given that $f(x) + f(-x) = 0$, confirming $f(x)$ is an odd function, the resulting definite integral $\int_a^x {f(t)dt}$ is determined to be an odd function based on its symmetry properties.

Therefore, $\int_a^x {f(t)dt}$ is an odd function.