Laplace Transform Definition & Basic Transforms

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The Laplace Transform is the tool that turns differential equations into algebra, and it is the foundation for four more parts in this series. Before you can solve a single circuit or mechanical system problem later in this series, you need the seven basic transform pairs memorized cold and three properties — linearity, the first shifting theorem, and change of scale — ready to apply without hesitation. This post covers the definition, the basic transform pairs, and these three properties for the ECE, EE, and ME board exams, with full step-by-step solutions for all 10 problems. This is Part 1 of the Laplace Transform Series.

📋 BOARD EXAM RELEVANCE

EE (Electrical Engineer) — Very high frequency. This is the foundation for every transient circuit analysis problem later in the series, and the basic pairs table is assumed knowledge on nearly every exam cycle.
ECE (Electronics Engineer) — Very high frequency. Signals and systems questions lean directly on the transform pairs and the shifting theorem, often without stating “Laplace Transform” explicitly in the question.
ME (Mechanical Engineer) — Moderate frequency. Vibration analysis and control system items reference Laplace-based transfer functions, which are built entirely from the pairs in this post.
CE (Civil Engineer) — Low frequency. Occasional structural dynamics or instrumentation-adjacent items reference the transform, but it is not a core CE board topic.
ChE, GeE, MetE, MinE, Naval Architect and Marine Engineer — Low frequency. Rarely tested directly; mostly appears in process control electives rather than the core board exam.

Bottom line: If you are taking the EE or ECE boards, treat the seven basic pairs in this post as non-negotiable memorization. Every other part in this series assumes instant recall of these, not re-derivation from the definition integral under exam time pressure.

The Definition

The Laplace Transform converts a function of time, $f(t)$ , into a function of a complex variable $s$ , denoted $F(s)$ . Board examiners rarely test the raw integral directly, but they test everything built from it, so understanding where the transform comes from matters even if you will not evaluate this integral by hand very often.

KEY FORMULA — THE LAPLACE TRANSFORM

$F(s) = \int_0^{\infty} f(t) e^{-st}\, dt$

Valid for $s > a$ , where $f(t)$ is piecewise continuous and of exponential order. In practice, you will use this integral only to derive the basic pairs below once — after that, every problem is a table lookup combined with the three properties in this post.

The Seven Basic Transform Pairs

These seven pairs are the entire foundation of the series. Memorize the pattern, not just the individual entries — notice that every trigonometric and hyperbolic pair has the same denominator shape, just with a sign difference between the circular and hyperbolic versions.

THE BASIC TRANSFORM PAIRS TABLE

$\mathcal{L}\{1\} = \dfrac{1}{s}, \quad s > 0$

$\mathcal{L}\{t^n\} = \dfrac{n!}{s^{n+1}}, \quad s > 0,\ n = 1, 2, 3, \ldots$

$\mathcal{L}\{e^{at}\} = \dfrac{1}{s-a}, \quad s > a$

$\mathcal{L}\{\sin(kt)\} = \dfrac{k}{s^2+k^2}, \quad s > 0$

$\mathcal{L}\{\cos(kt)\} = \dfrac{s}{s^2+k^2}, \quad s > 0$

$\mathcal{L}\{\sinh(kt)\} = \dfrac{k}{s^2-k^2}, \quad s > |k|$

$\mathcal{L}\{\cosh(kt)\} = \dfrac{s}{s^2-k^2}, \quad s > |k|$

Three Properties You Will Use Constantly

Nearly every problem past this point is one of these three properties applied to the basic pairs above.

LINEARITY

$\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$

The transform distributes over sums and scalar multiples. This alone lets you break any combination of the basic pairs into pieces you already know.

FIRST SHIFTING THEOREM (s-shift)

$\mathcal{L}\{e^{at}f(t)\} = F(s-a)$

Multiplying a function by $e^{at}$ in the time domain shifts its transform along the $s$ -axis. This is the shortcut behind almost every exponentially-modulated transform on the exam.

CHANGE OF SCALE

$\mathcal{L}\{f(at)\} = \dfrac{1}{a} F\left(\dfrac{s}{a}\right)$

Stretching or compressing the time axis scales the transform inversely. Less common on the boards than the two above, but it shows up in “find $F(s)$ given $f(2t)$ ” style items.

10 Worked Board Exam Problems

Problem 1. Find the Laplace Transform of a Combined Polynomial and Exponential Function

Problem: Find the Laplace Transform of $f(t) = 3e^{2t} + 4t^2 - 5\cos(3t)$ .

Given: $f(t) = 3e^{2t} + 4t^2 - 5\cos(3t)$

Find: $\mathcal{L}\{f(t)\}$

Solution:

Step 1: Apply linearity to split the expression into three separate transforms.

$\mathcal{L}\{f(t)\} = 3\mathcal{L}\{e^{2t}\} + 4\mathcal{L}\{t^2\} - 5\mathcal{L}\{\cos(3t)\}$

Step 2: Transform each piece using the basic pairs table.

$\mathcal{L}\{e^{2t}\} = \dfrac{1}{s-2}, \quad \mathcal{L}\{t^2\} = \dfrac{2!}{s^3} = \dfrac{2}{s^3}, \quad \mathcal{L}\{\cos(3t)\} = \dfrac{s}{s^2+9}$

Step 3: Multiply each transform back by its original coefficient and combine.

$F(s) = 3\left(\dfrac{1}{s-2}\right) + 4\left(\dfrac{2}{s^3}\right) - 5\left(\dfrac{s}{s^2+9}\right)$

✓ ANSWER: $F(s) = \dfrac{3}{s-2} + \dfrac{8}{s^3} - \dfrac{5s}{s^2+9}$

Examiner note: Linearity always comes first. Never try to transform a multi-term expression in one shot — split it, transform each piece independently, then recombine with the original coefficients.

Problem 2. Find the Laplace Transform of a Sine and Constant Combination

Problem: Find the Laplace Transform of $f(t) = 7\sin(4t) - 2$ .

Given: $f(t) = 7\sin(4t) - 2$

Find: $\mathcal{L}\{f(t)\}$

Solution:

Step 1: Split using linearity. Note that the constant $-2$ is really $-2 \cdot 1$ , so it uses the $\mathcal{L}\{1\}$ pair.

$\mathcal{L}\{f(t)\} = 7\mathcal{L}\{\sin(4t)\} - 2\mathcal{L}\{1\}$

Step 2: Apply the basic pairs, with $k = 4$ for the sine term.

$\mathcal{L}\{\sin(4t)\} = \dfrac{4}{s^2+16}, \quad \mathcal{L}\{1\} = \dfrac{1}{s}$

Step 3: Multiply by the original coefficients and combine.

$F(s) = 7\left(\dfrac{4}{s^2+16}\right) - 2\left(\dfrac{1}{s}\right)$

✓ ANSWER: $F(s) = \dfrac{28}{s^2+16} - \dfrac{2}{s}$

Examiner note: A bare constant is easy to miss as “just a number that doesn’t need transforming.” It does — it is $c \cdot 1$ , and $\mathcal{L}\{1\} = 1/s$ , not $1/s^0$ or anything else.

Problem 3. Apply the First Shifting Theorem to an Exponentially Modulated Cosine

Problem: Find the Laplace Transform of $f(t) = e^{3t}\cos(2t)$ .

Given: $f(t) = e^{3t}\cos(2t)$

Find: $\mathcal{L}\{f(t)\}$

Solution:

Step 1: Recognize the pattern $e^{at}g(t)$ with $a = 3$ and $g(t) = \cos(2t)$ . This calls for the first shifting theorem.

$\mathcal{L}\{e^{at}g(t)\} = G(s-a)$

Step 2: Find $G(s)$ , the transform of $g(t) = \cos(2t)$ alone, before shifting.

$G(s) = \mathcal{L}\{\cos(2t)\} = \dfrac{s}{s^2+4}$

Step 3: Apply the shift by replacing every $s$ in $G(s)$ with $(s-3)$ .

$F(s) = G(s-3) = \dfrac{s-3}{(s-3)^2+4}$

✓ ANSWER: $F(s) = \dfrac{s-3}{(s-3)^2+4}$

Examiner note: Do not expand $(s-3)^2$ in the final answer — board exams typically accept the shifted form as-is, and expanding it just introduces arithmetic risk for no benefit.

Problem 4. Find the Laplace Transform of Hyperbolic Functions

Problem: Find the Laplace Transform of $f(t) = 6\cosh(5t) - 3\sinh(5t)$ .

Given: $f(t) = 6\cosh(5t) - 3\sinh(5t)$

Find: $\mathcal{L}\{f(t)\}$

Solution:

Step 1: Apply linearity to separate the two hyperbolic terms.

$\mathcal{L}\{f(t)\} = 6\mathcal{L}\{\cosh(5t)\} - 3\mathcal{L}\{\sinh(5t)\}$

Step 2: Apply the hyperbolic pairs with $k = 5$ . Note both share the same denominator, $s^2 - 25$ .

$\mathcal{L}\{\cosh(5t)\} = \dfrac{s}{s^2-25}, \quad \mathcal{L}\{\sinh(5t)\} = \dfrac{5}{s^2-25}$

Step 3: Multiply by coefficients and combine over the common denominator.

$F(s) = 6\left(\dfrac{s}{s^2-25}\right) - 3\left(\dfrac{5}{s^2-25}\right) = \dfrac{6s - 15}{s^2-25}$

✓ ANSWER: $F(s) = \dfrac{6s-15}{s^2-25}$

Examiner note: Hyperbolic pairs look almost identical to the circular sine and cosine pairs — the only difference is the sign in the denominator. Mixing up $s^2+k^2$ and $s^2-k^2$ is one of the most common transcription errors on this topic.

Problem 5. Determine the Region of Convergence for a Combined Transform

Problem: For what values of $s$ does the Laplace Transform of $f(t) = e^{4t} + \cosh(6t)$ converge?

Given: $f(t) = e^{4t} + \cosh(6t)$

Find: The range of $s$ for which $\mathcal{L}\{f(t)\}$ exists.

Solution:

Step 1: Identify the convergence condition for each term separately.

$\mathcal{L}\{e^{4t}\}: \quad s > 4$

$\mathcal{L}\{\cosh(6t)\}: \quad s > |6| = 6$

Step 2: The combined transform only converges where every individual term converges — the stricter (larger) bound governs.

$s > 4 \quad \text{and} \quad s > 6 \implies s > 6$

✓ ANSWER: The transform converges for $s > 6$

Examiner note: When a function is a sum of terms with different convergence conditions, always take the strictest (largest) bound, not the loosest one. The transform is only valid where every term in the sum is valid.

Problem 6. Apply the Change of Scale Property

Problem: Given that $\mathcal{L}\{f(t)\} = \dfrac{1}{s^2+9}$ , find $\mathcal{L}\{f(2t)\}$ .

Given: $F(s) = \dfrac{1}{s^2+9}$

Find: $\mathcal{L}\{f(2t)\}$

Solution:

Step 1: Apply the change of scale property with $a = 2$ .

$\mathcal{L}\{f(at)\} = \dfrac{1}{a}F\left(\dfrac{s}{a}\right) \implies \mathcal{L}\{f(2t)\} = \dfrac{1}{2}F\left(\dfrac{s}{2}\right)$

Step 2: Substitute $s/2$ into $F(s)$ .

$F\left(\dfrac{s}{2}\right) = \dfrac{1}{(s/2)^2+9} = \dfrac{1}{\dfrac{s^2}{4}+9}$

Step 3: Simplify the compound fraction, then multiply by $1/2$ from the outer scaling factor.

$F\left(\dfrac{s}{2}\right) = \dfrac{4}{s^2+36} \implies \mathcal{L}\{f(2t)\} = \dfrac{1}{2}\left(\dfrac{4}{s^2+36}\right) = \dfrac{2}{s^2+36}$

✓ ANSWER: $\mathcal{L}\{f(2t)\} = \dfrac{2}{s^2+36}$

Examiner note: The $1/a$ factor sits outside the substituted function — do not forget to multiply by it after simplifying $F(s/a)$ . Dropping this factor is the single most common error on change-of-scale problems.

Problem 7. Find the Laplace Transform of a Higher-Degree Power Function

Problem: Find the Laplace Transform of $f(t) = 6t^4 - 3t + 8$ .

Given: $f(t) = 6t^4 - 3t + 8$

Find: $\mathcal{L}\{f(t)\}$

Solution:

Step 1: Apply linearity to split into three terms.

$\mathcal{L}\{f(t)\} = 6\mathcal{L}\{t^4\} - 3\mathcal{L}\{t\} + 8\mathcal{L}\{1\}$

Step 2: Apply the power function pair $\mathcal{L}\{t^n\} = n!/s^{n+1}$ for $n = 4$ and $n = 1$ .

$\mathcal{L}\{t^4\} = \dfrac{4!}{s^5} = \dfrac{24}{s^5}, \quad \mathcal{L}\{t\} = \dfrac{1!}{s^2} = \dfrac{1}{s^2}, \quad \mathcal{L}\{1\} = \dfrac{1}{s}$

Step 3: Multiply by coefficients and combine.

$F(s) = 6\left(\dfrac{24}{s^5}\right) - 3\left(\dfrac{1}{s^2}\right) + 8\left(\dfrac{1}{s}\right)$

✓ ANSWER: $F(s) = \dfrac{144}{s^5} - \dfrac{3}{s^2} + \dfrac{8}{s}$

Examiner note: $4! = 24$ , not $4 \times 3 = 12$ . Factorial errors on the power function pair are common under time pressure — write out the full factorial expansion if you are unsure.

Problem 8. Apply the First Shifting Theorem to a Shifted Sine Function

Problem: Find the Laplace Transform of $f(t) = e^{-2t}\sin(5t)$ .

Given: $f(t) = e^{-2t}\sin(5t)$

Find: $\mathcal{L}\{f(t)\}$

Solution:

Step 1: Match the pattern $e^{at}g(t)$ with $a = -2$ and $g(t) = \sin(5t)$ .

Step 2: Find $G(s)$ , the transform of $\sin(5t)$ alone.

$G(s) = \dfrac{5}{s^2+25}$

Step 3: Apply the shift, replacing $s$ with $(s-(-2)) = (s+2)$ .

$F(s) = G(s+2) = \dfrac{5}{(s+2)^2+25}$

✓ ANSWER: $F(s) = \dfrac{5}{(s+2)^2+25}$

Examiner note: When $a$ is negative, the shift becomes $s - (-2) = s+2$ . Watch the sign carefully — this is where most first shifting theorem mistakes happen on the boards.

Problem 9. Find the Original Function From a Shifted Transform

Problem: Given that $\mathcal{L}\{t^2\} = \dfrac{2}{s^3}$ , find $\mathcal{L}\{e^{5t}t^2\}$ .

Given: $\mathcal{L}\{t^2\} = \dfrac{2}{s^3}$

Find: $\mathcal{L}\{e^{5t}t^2\}$

Solution:

Step 1: Recognize this as the first shifting theorem with $a = 5$ and $g(t) = t^2$ , where $G(s)$ is already given.

$G(s) = \dfrac{2}{s^3}$

Step 2: Apply the shift by replacing every $s$ in $G(s)$ with $(s-5)$ .

$F(s) = G(s-5) = \dfrac{2}{(s-5)^3}$

✓ ANSWER: $F(s) = \dfrac{2}{(s-5)^3}$

Examiner note: This problem type hands you $G(s)$ directly to save you the lookup step, then tests whether you can apply the shift correctly. Do not re-derive $\mathcal{L}\{t^2\}$ from scratch when it is already given.

Problem 10. Find the Laplace Transform of a Voltage Signal in an EE Context

Problem: A voltage signal in a circuit is modeled by $v(t) = 12e^{-t}\cos(3t)$ volts for $t \geq 0$ . Find $\mathcal{L}\{v(t)\}$ .

Given: $v(t) = 12e^{-t}\cos(3t)$

Find: $\mathcal{L}\{v(t)\}$

Solution:

Step 1: Strip the physical units and treat this as a standard first shifting theorem problem with $a = -1$ and $g(t) = \cos(3t)$ .

Step 2: Find $G(s)$ , then apply linearity for the coefficient of 12.

$G(s) = \mathcal{L}\{\cos(3t)\} = \dfrac{s}{s^2+9}$

Step 3: Apply the shift, replacing $s$ with $(s+1)$ , then multiply by 12.

$F(s) = 12 \cdot G(s+1) = 12\left(\dfrac{s+1}{(s+1)^2+9}\right)$

✓ ANSWER: $F(s) = \dfrac{12(s+1)}{(s+1)^2+9}$

Examiner note: Board problems dressed up with physical context (voltage, current, displacement) reduce to the exact same math as the abstract versions. Strip the units, identify the pattern, apply the theorem — the physical story does not change the method.

Common Mistakes and Examiner Traps

❌ Common Mistake	✅ Correct Approach
Confusing the circular pairs ( $s^2+k^2$ ) with the hyperbolic pairs ( $s^2-k^2$ )	Sine and cosine use a plus sign in the denominator. Sinh and cosh use a minus sign. Double-check which family you are working with before writing the final answer.
Forgetting the $1/a$ scaling factor in the change of scale property	$\mathcal{L}\{f(at)\} = \dfrac{1}{a}F(s/a)$ . The $1/a$ sits outside the substituted function and is easy to drop under time pressure.
Applying the first shifting theorem with the wrong sign when $a$ is negative	$\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ always. If $a = -2$ , that becomes $F(s-(-2)) = F(s+2)$ , not $F(s-2)$ .
Miscalculating the factorial in the power function pair	$\mathcal{L}\{t^n\} = n!/s^{n+1}$ . Write out the factorial fully ( $4! = 24$ , not $12$ ) before substituting.
Treating a bare constant as not needing transformation	A constant $c$ is $c \cdot 1$ , and $\mathcal{L}\{1\} = 1/s$ . It still needs to go through the basic pairs table like every other term.
Using the loosest convergence bound instead of the strictest when combining terms	A sum of terms only converges where every term converges. Always take the largest (strictest) lower bound on $s$ among all terms in the sum.

Board Exam Quick Tips

Memorize the seven basic pairs cold. Nearly every later part in this series assumes instant recall of these, not re-derivation from the integral. Time spent re-deriving $\mathcal{L}\{t^n\}$ on exam day is time you do not have.
Linearity means you can always break a sum apart before transforming. Never try to transform a multi-term expression in one shot — split it into individual pieces first.
The first shifting theorem is just “replace $s$ with $s-a$ .” Find the transform of the un-shifted function first, then substitute. Watch the sign of $a$ carefully when it is negative.
Watch for sinh and cosh disguised as exponential combinations. They transform almost identically to sin and cos, just with a sign flip in the denominator — mixing these up is one of the most common point losses on this topic.
If a problem gives you $s > a$ as a condition, that number is not decoration. It can be the actual quantity being asked for, especially on convergence-classification items.

Frequently Asked Questions

Q1. Do I need to memorize the Laplace Transform integral itself for the board exam?

You need to recognize and understand it, but you will almost never evaluate $\int_0^{\infty} f(t)e^{-st}dt$ by hand on exam day. Board questions test whether you can apply the basic pairs and the three properties correctly, not whether you can re-derive the pairs from the integral under time pressure.

Q2. What is the difference between the first shifting theorem and the second shifting theorem?

The first shifting theorem, covered in this post, handles multiplication by $e^{at}$ in the time domain, which shifts $F(s)$ along the $s$ -axis. The second shifting theorem involves the unit step function and time-domain shifts, and is covered in Part 3 of this series when it becomes relevant to solving differential equations with switched inputs.

Q3. Why do sine and cosine use $s^2+k^2$ while sinh and cosh use $s^2-k^2$ ?

This traces back to the exponential definitions of these functions. $\sin(kt)$ and $\cos(kt)$ involve complex exponentials $e^{\pm ikt}$ , while $\sinh(kt)$ and $\cosh(kt)$ involve real exponentials $e^{\pm kt}$ . The sign difference in the denominator is a direct consequence of that underlying difference — it is not an arbitrary convention to memorize separately.

Q4. Can I apply the first shifting theorem and linearity in the same problem?

Yes, and this is common on the boards. Apply linearity first to split a multi-term expression apart, then apply the first shifting theorem to whichever individual terms have the $e^{at}$ pattern. The two properties are independent and stack cleanly.

Q5. What happens if a problem asks for the Laplace Transform of a function that is not in the basic pairs table?

Most functions that look unfamiliar can be rewritten using trigonometric or exponential identities until they match one of the seven basic pairs, or a combination of them under linearity. If a function genuinely cannot be reduced this way, it likely requires a technique from a later part in this series, such as the derivative property in Part 3 or the convolution theorem in Part 4.

What is Next

You now have the seven basic pairs and three properties that everything else in this series builds on. In Part 2, the direction reverses — instead of going from $f(t)$ to $F(s)$ , you will learn to go backward from $F(s)$ to $f(t)$ using partial fraction decomposition. This is where most board takers lose the most points in the entire Laplace Transform topic, so Part 2 spends most of its space on the three decomposition cases rather than the inverse transform definition itself.

→ Continue to Part 2 — Inverse Laplace Transform and Partial Fractions

→ Back to the Laplace Transform Series Index

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Laplace Transform Definition & Basic Transforms | Board Exam Reviewer

The Definition

The Seven Basic Transform Pairs

Three Properties You Will Use Constantly

10 Worked Board Exam Problems

Problem 1. Find the Laplace Transform of a Combined Polynomial and Exponential Function

Problem 2. Find the Laplace Transform of a Sine and Constant Combination

Problem 3. Apply the First Shifting Theorem to an Exponentially Modulated Cosine

Problem 4. Find the Laplace Transform of Hyperbolic Functions

Problem 5. Determine the Region of Convergence for a Combined Transform

Problem 6. Apply the Change of Scale Property

Problem 7. Find the Laplace Transform of a Higher-Degree Power Function

Problem 8. Apply the First Shifting Theorem to a Shifted Sine Function

Problem 9. Find the Original Function From a Shifted Transform

Problem 10. Find the Laplace Transform of a Voltage Signal in an EE Context

Common Mistakes and Examiner Traps

Board Exam Quick Tips

Frequently Asked Questions

What is Next

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The Definition

The Seven Basic Transform Pairs

Three Properties You Will Use Constantly

10 Worked Board Exam Problems

Problem 1. Find the Laplace Transform of a Combined Polynomial and Exponential Function

Problem 2. Find the Laplace Transform of a Sine and Constant Combination

Problem 3. Apply the First Shifting Theorem to an Exponentially Modulated Cosine

Problem 4. Find the Laplace Transform of Hyperbolic Functions

Problem 5. Determine the Region of Convergence for a Combined Transform

Problem 6. Apply the Change of Scale Property

Problem 7. Find the Laplace Transform of a Higher-Degree Power Function

Problem 8. Apply the First Shifting Theorem to a Shifted Sine Function

Problem 9. Find the Original Function From a Shifted Transform

Problem 10. Find the Laplace Transform of a Voltage Signal in an EE Context

Common Mistakes and Examiner Traps

Board Exam Quick Tips

Frequently Asked Questions

What is Next

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