Animated Solution for Mathematics - Definite Integration: If $g(x)={\int }_0^x\cos 4tdt$, then $g(x+\pi )$ equals

We are tasked with analyzing the function defined by the integral:

Think of this as an area function. The graph of $y=\cos (4t)$ is a wave oscillating with a frequency four times higher than the standard cosine wave. As $x$ increases, we accumulate the signed area under this wave.

A Step of Precision

To find the explicit form of $g(x)$, we apply the fundamental theorem of calculus. Using the rule $\int \cos (at)dt=\frac{\sin (at)}{a}$, where $a=4$, we perform the integration:

Applying the limits from $0$ to $x$, we obtain:

Since $\sin (0)=0$, the function simplifies to the elegant expression:

Exploring $g(x+\pi )$

Next, we investigate the behavior of $g(x+\pi )$ by substituting the shifted value into our derived function:

Recall that the sine function is periodic with a period of $2\pi$. Because $4\pi$ represents exactly two full cycles, the sine function remains unchanged:

Therefore, we conclude that:

To finalize our understanding, we evaluate $g(\pi )$ by substituting $\pi$ into our formula:

This result reveals a profound geometric truth: the integral of $\cos (4t)$ over the interval $[0,\pi ]$ covers exactly two full periods of the wave. Consequently, the positive and negative areas cancel out perfectly.

Given that $g(\pi )=0$, we observe that:

Both expressions are equivalent to $g(x)$, demonstrating the inherent symmetry and periodic nature of the function.