Studdy AI Logo
Math  /  Calculus

QuestionL{∫0tf(x)d}=\mathcal{L}\left\{\int_{0}^{t} f(x) d\right\}=

Studdy Solution

STEP 1

1. We are tasked with finding the Laplace transform of an integral.
2. The function f(x) f(x) is piecewise continuous on [0,∞) [0, \infty) .
3. The Laplace transform of a function f(t) f(t) is defined as L{f(t)}=∫0∞e−stf(t) dt \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt .

STEP 2

1. Recognize the Laplace transform of an integral.
2. Apply the Laplace transform to the integral.
3. Simplify the expression using known Laplace transform properties.

STEP 3

Recognize that we are dealing with the Laplace transform of an integral of the form:
L{∫0tf(x) dx} \mathcal{L}\left\{\int_{0}^{t} f(x) \, dx \right\}
This is a standard form that can be solved using the property of the Laplace transform related to integrals.

STEP 4

Apply the Laplace transform to the integral. According to the property of the Laplace transform for integrals, we have:
L{∫0tf(x) dx}=1sL{f(t)} \mathcal{L}\left\{\int_{0}^{t} f(x) \, dx \right\} = \frac{1}{s} \mathcal{L}\{f(t)\}
This property states that the Laplace transform of an integral from 0 to t t of a function f(x) f(x) is equal to 1s \frac{1}{s} times the Laplace transform of the function f(t) f(t) .

STEP 5

Simplify the expression using the known property:
L{∫0tf(x) dx}=1sF(s) \mathcal{L}\left\{\int_{0}^{t} f(x) \, dx \right\} = \frac{1}{s} F(s)
where F(s)=L{f(t)} F(s) = \mathcal{L}\{f(t)\} .
The Laplace transform of the integral is:
1sF(s) \boxed{\frac{1}{s} F(s)}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord