Question

Ind the laplace transform without using the chart (घ) use L[∫ᵗ₀

g(u)du]=L[g]/s f(t)=t∫ᵗ₀ ​e⁻ᵘ u² du

Answer 1

To find the Laplace transform of the given function, apply the formula L[∫ᵗ₀ g(u)du]=L[g]/s and simplify the integral.

Step-by-step explanation:

To find the Laplace transform of the given function, we can use the formula L[∫ᵗ₀ g(u)du]=L[g]/s. In this case, the function is f(t)=t∫ᵗ₀ ​e⁻ᵘ u² du. Let's find the Laplace transform by applying the formula and simplifying the integral:

1. Apply the formula: L[f(t)] = L[t∫ᵗ₀ ​e⁻ᵘ u² du] = 1/s * L[∫ᵗ₀ ​e⁻ᵘ u² du]
2. Simplify the integral: L[∫ᵗ₀ g(u)du] = L[g]/s = 1/s * L[e⁻ᵘ u² du]
3. Apply the Laplace transform to the integral: L[e⁻ᵘ u² du] = 1/(s+1)³
4. Substitute the result back into the equation: L[f(t)] = 1/s * 1/(s+1)³

So, the Laplace transform of the given function is 1/s * 1/(s+1)³.