Question

Assuming \( s>|a| \); the Laplace transform of \( f(x) = \cosh(ax) \) is:

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

Hint:

Always remember key Laplace transform pairs for hyperbolic functions:
- \( \mathcal{L}[\cosh(ax)] = \frac{s}{s^2 - a^2} \)
- \( \mathcal{L}[\sinh(ax)] = \frac{a}{s^2 - a^2} \)
Applicable when \( s>|a| \).

Answer 1

The Correct Option is C

Step 1: Recall the Laplace transform definition for hyperbolic cosine.
The Laplace transform of \( \cosh(ax) \) is given by the standard formula:
\[ \mathcal{L}[\cosh(ax)] = \int_0^\infty e^{-sx} \cosh(ax) \, dx \]
Step 2: Use the known Laplace transform identity:
\[ \mathcal{L}[\cosh(ax)] = \frac{s}{s^2 - a^2}, \quad \text{for } s > |a| \]
Step 3: Match with given options.
Only option (C) matches the correct transform result.