1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
tatiyna
3 years ago
12

What is the Laplace Transform of 7t^3 using the definition (and not the shortcut method)

Mathematics
1 answer:
Leokris [45]3 years ago
5 0

Answer:

Step-by-step explanation:

By definition of Laplace transform we have

L{f(t)} = L{{f(t)}}=\int_{0}^{\infty }e^{-st}f(t)dt\\\\Given\\f(t)=7t^{3}\\\\\therefore L[7t^{3}]=\int_{0}^{\infty }e^{-st}7t^{3}dt\\\\

Now to solve the integral on the right hand side we shall use Integration by parts Taking 7t^{3} as first function thus we have

\int_{0}^{\infty }e^{-st}7t^{3}dt=7\int_{0}^{\infty }e^{-st}t^{3}dt\\\\= [t^3\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(3t^2)\int e^{-st}dt]dt\\\\=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\

Again repeating the same procedure we get

=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt= \frac{3}{s}[t^2\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t^2)\int e^{-st}dt]dt\\\\=\frac{3}{s}[0-\int_{0}^{\infty }\frac{2t^{1}}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{2}}[\int_{0}^{\infty }te^{-st}dt]\\\\

Again repeating the same procedure we get

\frac{3\times 2}{s^2}[\int_{0}^{\infty }te^{-st}dt]= \frac{3\times 2}{s^{2}}[t\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t)\int e^{-st}dt]dt\\\\=\frac{3\times 2}{s^2}[0-\int_{0}^{\infty }\frac{1}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{3}}[\int_{0}^{\infty }e^{-st}dt]\\\\

Now solving this integral we have

\int_{0}^{\infty }e^{-st}dt=\frac{1}{-s}[\frac{1}{e^\infty }-\frac{1}{1}]\\\\\int_{0}^{\infty }e^{-st}dt=\frac{1}{s}

Thus we have

L[7t^{3}]=\frac{7\times 3\times 2}{s^4}

where s is any complex parameter

You might be interested in
Alicia is writing the program for a video game. For one part of the game she uses the rule ( x , y) ----> ( x - 3, y + 4) to
anastassius [24]
The "rule" being described here is nothing more than the input/output of a mathematical function. 

<span>For every input 'x' value supplied, you only need to subtract three to it. For every input 'y' value, you only need to add four to it. </span>

<span>Example: I'll use variable 'm' to represent this function. Variable 'p' will represent the current input point. </span>

<span>m(p) = p[x - 3, y + 4] = p[-7 - 3, 0 + 4] = p[-10, 4]. 'p[]" is just the point.</span>
5 0
3 years ago
Can some o e help me with 7 please this is due tommorow
erik [133]
This is kinda hard to explain but I'll try my best. Here's a tip, the fractions that are on the ruler really mean 10,20,30,40,50,60,70,80,90, and 100. So, when you look at the decimals on number 7, you have to try to turn them into fractions in your head. For example, 0.9 is actually 9/10 or 90.
8 0
3 years ago
Read 2 more answers
Under what circumstances are two nonright triangles congruent?
soldier1979 [14.2K]
If these are the given choices of the above problem,

a. one side and one angle are equal. 
<span>b.three sides are equal </span>
<span>c.two angles are equal </span>
<span>d. three angles are equal

Two non-right triangles are congruent when B. THREE SIDES ARE EQUAL.

Two triangles are congruent if:
1) All corresponding sides are equal (SSS)
2) A pair of corresponding sides and the included angle are equal (SAS)
3) A pair of corresponding angles and the included side are equal (ASA)
4) A pair of corresponding angles and a non-included side are equal (AAS)</span>
3 0
3 years ago
What is 24/7 as a mixed number
Masteriza [31]
I can't do the fraction bar but it's 3 and 3/7
6 0
3 years ago
Read 2 more answers
The answer how you set it up
Hunter-Best [27]
You sign up by getting a account then answering other peoples questions.
6 0
3 years ago
Other questions:
  • If 4 gallons of milk costs $16.76, how much would 7 gallons of milk cost?
    14·2 answers
  • Jenny is cooking a chicken for dinner. A chicken needs to be cooked for 40 mins per kilogram, plus an extra 20 mins. How long wi
    12·1 answer
  • A bicycle’s wheel has a radius of 32.5 cm. What distance will the bicycle travel if the wheel turns 100,000 times?
    10·2 answers
  • What is the greatest common factor of 35y4, 14y4, and 63y4?
    7·1 answer
  • Classify the real number(natural,whole,integer,rational,irrational).
    11·1 answer
  • How many complex roots does the polynomial equation below have?
    10·1 answer
  • Joey spins a spinner and lands on the number 4. Then Jennifer spins and lands on the number 2. Are these events independent or d
    8·1 answer
  • A regular pentagon ABCDE is shown work out angle x
    5·1 answer
  • Emma starts at one end of a 45-mile-long bike trail. After
    8·1 answer
  • SOMEONE PLS HELP WHAT DO I DO HERE
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!