Question

The Laplace transform of the multiple integral \( L \left\{ \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} \int_{0}^{t} \cos au \, du \, du \, du \, du \right\} \) is:

• A. \( \frac{1}{s^2 + a^2} \)
• B. \( \frac{1}{s(s^2 + a^2)} \)
• C. \( \frac{1}{s^2(s^2 + a^2)} \)
• D. \( \frac{1}{s^3(s^2 + a^2)} \)

Hint:

Always reduce the power of \( s \) in the numerator by 1 for every level of integration when dealing with functions like \( \cos at \) or \( \sin at \).

Answer 1

The Correct Option is D

Concept: The Laplace transform of an integral of a function is a standard operational property.

• If \( L\{f(t)\} = F(s) \), then \( L\left\{ \int_{0}^{t} f(u) du \right\} = \frac{1}{s} F(s) \).

• For \( n \) repeated integrals, the transform becomes \( \frac{1}{s^n} F(s) \).

• The Laplace transform of \( \cos at \) is \( \frac{s}{s^2 + a^2} \).

Step 1: Find the Laplace transform of the integrand.
Let the inner function be \( f(t) = \cos at \). The Laplace transform is: \[ L\{\cos at\} = \frac{s}{s^2 + a^2} \]

Step 2: Identify the number of integrations.
The expression involves four nested integrals. According to the property of integrals in Laplace transforms: \[ L\left\{ \int_{0}^{t} \dots \int_{0}^{t} f(u) (du)^4 \right\} = \frac{1}{s^4} \cdot L\{f(t)\} \]

Step 3: Calculate the final transform.
Substitute the transform of the cosine function. \[ \text{Result} = \frac{1}{s^4} \cdot \left( \frac{s}{s^2 + a^2} \right) \] Simplify the powers of \( s \). \[ \text{Result} = \frac{1}{s^3(s^2 + a^2)} \] Final Answer: (D)