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
Alinara [238K]
2 years ago
15

Evaluate using integration by parts ​

Mathematics
1 answer:
PolarNik [594]2 years ago
8 0

Rather than carrying out IBP several times, let's establish a more general result. Let

I(n)=\displaystyle\int x^ne^x\,\mathrm dx

One round of IBP, setting

u=x^n\implies\mathrm du=nx^{n-1}\,\mathrm dx

\mathrm dv=e^x\,\mathrm dx\implies v=e^x

gives

\displaystyle I(n)=x^ne^x-n\int x^{n-1}e^x\,\mathrm dx

I(n)=x^ne^x-nI(n-1)

This is called a power-reduction formula. We could try solving for I(n) explicitly, but no need. n=5 is small enough to just expand I(5) as much as we need to.

I(5)=x^5e^x-5I(4)

I(5)=x^5e^x-5(x^4e^x-4I(3))=(x-5)x^4e^x+20I(3)

I(5)=(x-5)x^4e^x+20(x^3e^x-3I(2))=(x^2-5x+20)x^3e^x-60I(2)

I(5)=(x^2-5x+20)x^3e^x-60(x^2e^x-2I(1))=(x^3-5x^2+20x-60)x^2e^x+120I(1)

I(5)=(x^3-5x^2+20x-60)x^2e^x+120(xe^x-I(0))

Finally,

I(0)=\displaystyle\int e^x\,\mathrm dx=e^x+C

so we end up with

I(5)=(x^4-5x^3+20x^2-60x+120)xe^x-120e^x+C

I(5)=(x^5-5x^4+20x^3-60x^2+120x-120)e^x+C

and the antiderivative is

\displaystyle\int2x^5e^x\,\mathrm dx=(2x^5-10x^4+40x^3-120x^2+240x-240)e^x+C

You might be interested in
1) WET SURPRISE A water balloon is launched off of a patio. The path of the water
mart [117]

The function of the water balloon is an illustration of a quadratic function.

The balloon will splat on the ground after 2.5 seconds

The function is given as:

h(t) = -6t^2 + 13t + 5

When the balloon splats on the ground, the value of h(t) is 0.

So, we have:

-6t^2 + 13t + 5 = 0

Expand the expression

-6t^2 + 15t- 2t + 5 = 0

Factorize the above expression

-3t(2t - 5)- 1(2t - 5 )= 0

Factor out 2t - 5 in the 2t - 5

(-3t - 1)(2t - 5 )= 0

Split the equation into 2

-3t - 1 = 0\ or\ 2t - 5= 0

Rewrite as:

-3t = 1\ or\ 2t = 5

Solve for t

t = -\frac 13\ or\ t = \frac 52

The value of t cannot be less than 0 i.e. negative.

So, we have

t = \frac 52

Divide 5 by 2

t = 2.5

Hence, the balloon will splat on the ground after 2.5 seconds

Read more about quadratic functions at:

brainly.com/question/11631534

6 0
2 years ago
Given m||n, find the value of x.
USPshnik [31]

Answer:

57 degrees

Step-by-step explanation:

Same side interior angles are supplementary which means they both equal 180 degrees.

3 0
3 years ago
Type the correct answer in each box. Write your answers in decimal form, rounded to the nearest tenth, if necessary. Type the so
strojnjashka [21]

Answer:

-1/2 and -1

Step-by-step explanation:

This is the answer I used in my test, not sure if it is correct, but hope it helps!

8 0
2 years ago
What is the slope of a line with a x-intercept of -3 and a y-intercept of -4?
GuDViN [60]

The x intercept is -3 so the point (-3,0) is on the line. This point is on the horizontal x axis. The x intercept is always in the form (NUMBER,0) where you replace "NUMBER" with any number you want.

The y intercept is -4 so we know that (0,-4) is also on the line. This is on the vertical y axis. The y intercept is always in the form (0,NUMBER)

Use the slope formula to find the slope of the line through the points (-3,0) and (0,-4)

m = (y2-y1)/(x2-x1)

m = (-4-0)/(0-(-3))

m = (-4-0)/(0+3)

m = -4/3

The answer is choice B: -4/3

I have a feeling you mixed up x and y somehow when you got -3/4

6 0
3 years ago
Write an inequality to represent the situation the temperature stays above –15°
lozanna [386]
Let t = the temperature
stays above → > also above tells you it is not equal to the given number

Answer t > -15
6 0
3 years ago
Other questions:
  • The US courts are reviewing a case to see whether a work has been refused fairly. What is an example of something they might con
    12·1 answer
  • If 80% of scheduled flights actually take place and cancellations are independent events, what is the probability that 3 separat
    11·1 answer
  • How would -9-(-5) look like as an addition problem?
    7·2 answers
  • Select the correct answer. During the summer, Jody earns $10 per hour babysitting and $15 per hour doing yardwork. This week she
    10·1 answer
  • A community is planning to build a rectangular garden.the width of the garden is 27/4 feet and the perimeter of the garden is 37
    5·1 answer
  • Pls help will mark brainliest
    6·2 answers
  • Rewrite the following expression using only addition: 4x-2y-8
    11·1 answer
  • 1 & 1/3 plus 1 & 3/4
    12·2 answers
  • HELP ME PLEASE..!!!!!! Thank youuuuuu…
    15·1 answer
  • The period 7 (in seconds) of a pendulum is given by T = 2pi√( L divided by 32) where L stands for the length (in feet) of the pe
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!