Math Snap

STEP 1

1. We are given the Laplace transform $F(s) = \ln \left(1+\frac{1}{s^{2}}\right)$.
2. We need to find the inverse Laplace transform of $F(s)$.
3. We are given four possible options for the inverse Laplace transform.

STEP 2

1. Recognize the form of the given Laplace transform.
2. Use known Laplace transform properties or theorems to simplify.
3. Identify the inverse Laplace transform from the given options.

STEP 3

Recognize that $F(s) = \ln \left(1+\frac{1}{s^{2}}\right)$ can be related to the Laplace transform of a known function. The function inside the logarithm suggests a series expansion approach.

STEP 4

Recall the series expansion for $\ln(1+x)$:
$$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$$ Apply this to $\ln \left(1+\frac{1}{s^{2}}\right)$:
$$\ln \left(1+\frac{1}{s^{2}}\right) = \frac{1}{s^2} - \frac{1}{2s^4} + \frac{1}{3s^6} - \cdots$$

STEP 5

Recognize that each term in the series corresponds to a known Laplace transform:
- The inverse Laplace transform of $\frac{1}{s^2}$ is $t$.
- The inverse Laplace transform of $\frac{1}{s^4}$ is $\frac{t^3}{6}$.
- The inverse Laplace transform of $\frac{1}{s^6}$ is $\frac{t^5}{120}$.
Thus, the inverse Laplace transform of the series is:
$$t - \frac{t^3}{2 \cdot 6} + \frac{t^5}{3 \cdot 120} - \cdots$$

STEP 6

Recognize that the series represents the function:
$$\frac{\sin(t)}{t}$$ This is because the series expansion of $\frac{\sin(t)}{t}$ is:
$$\frac{\sin(t)}{t} = 1 - \frac{t^2}{3!} + \frac{t^4}{5!} - \cdots$$

Identify the correct option from the given choices. The inverse Laplace transform is:
b. $\frac{\sin(t)}{t}$
The inverse Laplace transform of $F(s) = \ln \left(1+\frac{1}{s^{2}}\right)$ is:
$$\boxed{\frac{\sin(t)}{t}}$$