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
kaheart [24]
3 years ago
12

This is a question on my partial fractions homework, but no matter what I try I can't figure it out..

Mathematics
1 answer:
Ierofanga [76]3 years ago
4 0
\dfrac{x^2+x+1}{(x+1)^2(x+2)}=\dfrac{a_1x+a_0}{(x+1)^2}+\dfrac b{x+2}
\implies\dfrac{x^2+x+1}{(x+1)^2(x+2)}=\dfrac{(a_1x+a_0)(x+2)+b(x+1)^2}{(x+1)^2(x+2)}
\implies x^2+x+1=(a_1+b)x^2+(2a_1+a_0+2b)x+(2a_0+b)
\implies\begin{cases}a_1+b=1\\2a_1+a_0+2b=1\\2a_0+b=1\end{cases}\implies a_1=-2,a_0=-1,b=3

So you have

\displaystyle\int_0^2\frac{x^2+x+1}{(x+1)^2(x+2)}\,\mathrm dx=-2\int_0^2\frac x{(x+1)^2}\,\mathrm dx-\int_0^2\frac{\mathrm dx}{(x+1)^2}+3\int_0^2\frac{\mathrm dx}{x+2}
=\displaystyle-2\int_1^3\dfrac{x-1}{x^2}\,\mathrm dx-\int_0^2\frac{\mathrm dx}{(x+1)^2}+3\int_0^2\frac{\mathrm dx}{x+2}

where in the first integral we substitute x\mapsto x+1.

=\displaystyle-2\int_1^3\left(\frac1x-\frac1{x^2}\right)\,\mathrm dx-\frac1{1+x}\bigg|_{x=0}^{x=2}+3\ln|x+2|\bigg|_{x=0}^{x=2}
=-2\left(\ln|x|+\dfrac1x\right)\bigg|_{x=1}^{x=3}-\dfrac23+3(\ln4-\ln2)
=-2\left(\ln3+\dfrac13-1\right)-\dfrac23+3\ln2
=\dfrac23+\ln\dfrac89
You might be interested in
PLEASE HELP WITH THIS MATH
Anni [7]

Answer:

Step-by-step explanation:

1. 96 centimeters squared

2. 300 degrees

3. no, the diagonals of a parallelogram donor intersect at 90 degree ( are not perpendicular) .it is because the adjacent sides of parallelogram are not equal..

4. no

4 0
2 years ago
How to solve -3 1/2 + 5 4/5?
monitta
Hi. This is a bit lengthy so I am attaching an image that shows it worked out step by step. I hope it helps.

Take care,
Diana

6 0
3 years ago
I need help with this problem #8 ignore the scribbles
devlian [24]
There is no picture that I see:/
4 0
3 years ago
(4÷y+2) +(2÷y-3) =(6÷y+6)
Evgesh-ka [11]
<span>Simplifying 4(y + -3) = 6(y + 2) Reorder the terms: 4(-3 + y) = 6(y + 2) (-3 * 4 + y * 4) = 6(y + 2) (-12 + 4y) = 6(y + 2) Reorder the terms: -12 + 4y = 6(2 + y) -12 + 4y = (2 * 6 + y * 6) -12 + 4y = (12 + 6y) Solving -12 + 4y = 12 + 6y Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6y' to each side of the equation. -12 + 4y + -6y = 12 + 6y + -6y Combine like terms: 4y + -6y = -2y -12 + -2y = 12 + 6y + -6y Combine like terms: 6y + -6y = 0 -12 + -2y = 12 + 0 -12 + -2y = 12 Add '12' to each side of the equation. -12 + 12 + -2y = 12 + 12 Combine like terms: -12 + 12 = 0 0 + -2y = 12 + 12 -2y = 12 + 12 Combine like terms: 12 + 12 = 24 -2y = 24 Divide each side by '-2'. y = -12 Simplifying y = -12</span>
3 0
3 years ago
PLEASE HELP ASAP!! WILL GIVE POINTS AND MARK BRAINLIEST!!!!!
Korvikt [17]

meme break

⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⣤⣶⣶ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣾⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⡏⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿ ⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⣿ ⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠙⠿⠿⠿⠻⠿⠿⠟⠿⠛⠉⠀⠀⠀⠀⠀⣸⣿ ⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢰⣹⡆⠀⠀⠀⠀⠀⠀⣭⣷⠀⠀⠀⠸⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠉⠀⠀⠤⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣷⠀⠀⠀⠀⡠⠤⢄⠀⠀⠀⠠⣿⣿⣷⠀⢸⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⡀⠉⠀⠀⠀⠀⠀⢄⠀⢀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿

6 0
3 years ago
Other questions:
  • Find the inverse of the following functions f(x)=6-3x g(x)=x+2
    11·2 answers
  • A manufacturing company with 450 employees has a drop a product line and now only has 420 employees what is the percent of decre
    7·1 answer
  • Please answer and explain!!!
    6·1 answer
  • Help pleaasssseeee (use picture) it’ll only take some time out of your day but will be a big impact on my day thank you hope you
    13·1 answer
  • Plz help me ................
    13·1 answer
  • write the equation of the line that passes through the point (-2,4) and (5,8). Put your answer in fully reduced point slope form
    11·1 answer
  • HELP FAST!!
    15·1 answer
  • Can some PLEASE help me this is my 3rd time asking the same question
    12·1 answer
  • PLEASE HELP!!!!! DUE IN 5!!!!
    5·1 answer
  • Scarlett is buying treats from a candy shop to share with her friends from Science Club. There's a Buggy Bundle box with 8 butte
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!