---
title: "Energy and power of 2e^(−5t)u(t−1)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-2e-5t"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-2e-5t
site: BCT Engineering
---

# Energy and power of 2e^(−5t)u(t−1)

Signals and Systems · Chapter 1: Signals

- Asked in: 2082 Bhadra (new course), 4 marks
- Video: [Energy and Power Signals | Solved Problems / Examples](https://www.youtube.com/watch?v=bZgQD_7d5VM) — Shrenik Jain
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-2e-5t
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Calculate the energy and power of the signal x(t) = 2e^(−5t)u(t−1).

## Short answer

Energy is computed by integrating |x(t)|^2 = 4e^(-10t) from t=1 to ∞: E = 0.4e^(-10) ≈1.816×10^(-5) J. Power is the limit of average power over infinite interval, P = 0. Finite energy and zero power make it an energy signal.

## Answer

The signal $x(t) = 2e^{-5t}u(t-1)$ is zero for $t < 1$ and decays exponentially for $t \ge 1$.

**Energy:**

$$
\begin{aligned}
E &= \int_{-\infty}^{\infty}|x(t)|^2\,dt = \int_{1}^{\infty}4e^{-10t}\,dt \\
&= 4\left[\frac{e^{-10t}}{-10}\right]_1^{\infty} = 4\left(0 + \frac{e^{-10}}{10}\right) \\
&= 0.4\,e^{-10} = 1.816\times 10^{-5}\ \text{J}
\end{aligned}
$$

**Power:**

$$
P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2dt = \lim_{T\to\infty}\frac{1}{2T}\cdot 0.4\left(e^{-10} - e^{-10T}\right) = 0
$$

**Answer:** $E = 0.4e^{-10} \approx 1.816\times10^{-5}$ J and $P = 0$. The energy is finite, so $x(t)$ is an **energy signal**.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For x(t) = 2e^(-5t)u(t-1), what is the energy integral used?
   A. integral from 0 to infinity of 2e^(-5t) dt
   B. integral from 1 to infinity of 4e^(-10t) dt
   C. integral from 1 to infinity of 2e^(-10t) dt
   D. integral from -infinity to infinity of 4e^(-5t) dt
   Answer: B. Energy uses |x|^2 = 4e^(-10t), and u(t-1) restricts the integral to t >= 1. (Key point: Energy integral limits)

2. What is the energy of the signal?
   A. 0.4e^(-10) = 1.816x10^-5
   B. 4e^(-10)
   C. 0.4e^(-5)
   D. 0.2e^(-10)
   Answer: A. 4 x e^(-10)/10 = 0.4e^(-10), about 1.816x10^-5 J. (Key point: Energy value)

3. What is the average power and why?
   A. Equal to the energy, since E is small
   B. Infinite, since the signal never ends
   C. 0, since finite energy is divided by an infinite time 2T
   D. 0.4, the coefficient of the integral
   Answer: C. P is the limit of E-like finite integral divided by 2T as T goes to infinity, which tends to 0. (Key point: Power of energy signal)

4. How is the signal classified?
   A. Power signal
   B. Energy signal
   C. Both energy and power signal
   D. Neither energy nor power signal
   Answer: B. Finite energy and zero power make it an energy signal. (Key point: Signal classification)

## Related in Signals and Systems

- [Energy or power: e^(−5t)u(t+2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-shifted-exponential) — Signals and Systems, chapter 1, asked 5×
- [Power of energy signal is zero](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-signal-example) — Signals and Systems, chapter 1
- [Energy and power of (3+4j)e^(2t)u(−t)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-complex-exponential) — Signals and Systems, chapter 1
- [Energy or power: x(t) = e^(−at)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-one-sided-exponential) — Signals and Systems, chapter 1, asked 6×
- [Energy or power: x(t) = e^(−3t)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-decaying-exponential) — Signals and Systems, chapter 1, asked 5×
- [Energy or power: x(t) = e^(−a|t|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-two-sided-exponential) — Signals and Systems, chapter 1, asked 2×

## Related in other subjects

- [Energy and power signal; bandwidth of a system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenergy-power-bandwidth-system) — Communication Systems, chapter 1
- [Rayleigh energy theorem; power from Fourier series](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arayleigh-periodic-power-fourier) — Communication Systems, chapter 1
- [ESD of an energy signal (ideal BPF example)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-energy-signal-ideal-bpf) — Communication Systems, chapter 1
- [Energy or power: x(t) = A cos2πft](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenergy-power-acos) — Communication Systems, chapter 1, asked 2×