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
djyliett [7]
2 years ago
5

Integral of x"2+4/x"2+4x+3

Mathematics
1 answer:
dolphi86 [110]2 years ago
3 0

I'm guessing you mean

\displaystyle \int\frac{x^2+4}{x^2+4x+3}\,\mathrm dx

First, compute the quotient:

\displaystyle \frac{x^2+4}{x^2+4x+3} = 1 + \frac{4x-1}{x^2+4x+3}

Split up the remainder term into partial fractions. Notice that

<em>x</em> ² + 4<em>x</em> + 3 = (<em>x</em> + 3) (<em>x</em> + 1)

Then

\displaystyle \frac{4x-1}{x^2+4x+3} = \frac a{x+3} + \frac b{x+1} \\\\ \implies 4x - 1 = a(x+1) + b(x+3) = (a+b)x + a+3b \\\\ \implies a+b=4 \text{ and }a+3b = -1 \\\\ \implies a=\frac{13}2\text{ and }b=-\frac52

So the integral becomes

\displaystyle \int \left(1 + \frac{13}{2(x+3)} - \frac{5}{2(x+1)}\right) \,\mathrm dx = \boxed{x + \frac{13}2\ln|x+3| - \frac52 \ln|x+1| + C}

We can simplify the result somewhat:

\displaystyle x + \frac{13}2\ln|x+3| - \frac52 \ln|x+1| + C \\\\ = x + \frac12 \left(13\ln|x+3| - 5\ln|x+1|\right) + C \\\\ = x + \frac12 \left(\ln\left|(x+3)^{13}\right| - \ln\left|(x+1)^5\right|\right) + C \\\\ = x + \frac12 \ln\left|\frac{(x+3)^{13}}{(x+1)^5}\right| + C \\\\ = \boxed{x + \ln\sqrt{\left|\frac{(x+3)^{13}}{(x+1)^5}\right|} + C}

You might be interested in
Help people please i have like 3 minutes left
Serjik [45]

Answer:

b

Step-by-step explanation:

4 0
3 years ago
Given that the line y=ax+b passes through the points A(0; 3) and B(-2;7), how do I find the values of a and b?
Andru [333]
First you have find the slope (a) in order to find the value of the y intercept (b).

To compute the slope, you simply have to find the difference of y coordinates and divide them by the difference of x coordinates:

a = (y2-y1)/(x2-x1)
a = (7-3)/(-2–0)
a = 4/-2
a = -2

To compute the y intercept denoted by b, we substitute the value of a with -2 in the formula and choose any point (x,y) and substitute their values in the formula as well. Like so:

I choose point A:

3 = (-2)(0) + b
3 = b

4 0
2 years ago
Three-fifths of the members of the Spanish club are girls. There are a total of 30 girls in the Spanish club.
erastova [34]

Problem: You do not have an attached picture. Though, I can help you out with this question.

So, three-fifths of the members are girls. There are 30 of girls. We can immediately make an equation using "x".

3/5x=30

3/5x × 5/3=30 × 5/3

x=50

Thus, there are 50 members in total. Though if x is too hard for you, you may substitute it by using another variable.

Variable in here is x.

3 0
2 years ago
A chemical company is considering two processes
zhannawk [14.2K]

Answer: $166.430 per year

Step-by-step explanation:

0= -900000+AOC(P/A, 40%,3)

0= -900000+AOC(1.5889)

result we get

AOC= $566430

Require reduction = 566430-400000

result we get

$166.430 per year

8 0
3 years ago
HELP ASAP WILL GIVE BRAINLIEST IF CORRECT
Anna35 [415]
The expression that can be used is 3.14 (6x4+6^2)

The surface area is 188.4
5 0
3 years ago
Read 2 more answers
Other questions:
  • please help :) Respecting and understanding different opinions promotes fill in the blank . answer choices : (value/tolerance)
    5·1 answer
  • We can describe 3x-2 as an expression how can we describe the parts of the expression that the arrows point to?
    9·1 answer
  • OFFERING A LOT OF POINTS. please write 2 or
    8·1 answer
  • Anybody can help ?? 6-3d=5(2-d)
    8·2 answers
  • A local little league has a total of 95 players, of whom 40% are right handed. How many right handed players are there?
    11·2 answers
  • An internet search engine looks for a certain keyword in a sequence of independent web sites. It is believed that 20% of the sit
    7·1 answer
  • A college professor's compensation package includes the total cost of a $350-per-month health insurance plan, the total cost of
    9·2 answers
  • Pls help!!!!!!!!!!!!!!!!!
    13·1 answer
  • What is the value of the expression below when y==8?<br><img src="https://tex.z-dn.net/?f=y%20%20-%20y%20-%202" id="TexFormula1"
    6·1 answer
  • What is the lowest common multiple of 24, 8 and 16.
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!