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
nadya68 [22]
3 years ago
11

Find the integral using substitution or a formula.

Mathematics
1 answer:
Nadusha1986 [10]3 years ago
8 0
\rm \int \dfrac{x^2+7}{x^2+2x+5}~dx

Derivative of the denominator:
\rm (x^2+2x+5)'=2x+2

Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.

\rm \int \dfrac{x^2+2+5}{x^2+2x+5}~dx

and then split the fraction,

\rm \int \dfrac{x^2+2}{x^2+2x+5}~dx+\int\dfrac{5}{x^2+2x+5}~dx

Based on our previous test, we know that a simple substitution will work for the first integral: \rm \quad u=x^2+2x+5\qquad\to\qquad du=2x+2~dx

So the first integral changes,

\rm \int \dfrac{1}{u}~du+\int\dfrac{5}{x^2+2x+5}~dx

integrating to a log,

\rm ln|x^2+2x+5|+\int\dfrac{5}{x^2+2x+5}~dx

Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.

\rm x^2+2x+5=(x^2+2x+1)+4=(x+1)^2+2^2

So we have this going on,

\rm ln|x^2+2x+5|+\int\dfrac{5}{(x+1)^2+2^2}~dx

Let's factor the 5 out of the intergral,
and the 4 from the denominator,

\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\frac{(x+1)^2}{2^2}+1}~dx

Bringing all that stuff together as a single square,

\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(\dfrac{x+1}{2}\right)^2+1}~dx

Making the substitution: \rm \quad u=\dfrac{x+1}{2}\qquad\to\qquad 2du=dx

giving us,

\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(u\right)^2+1}~2du

simplying a lil bit,

\rm ln|x^2+2x+5|+\frac52\int\dfrac{1}{u^2+1}~du

and hopefully from this point you recognize your arctangent integral,

\rm ln|x^2+2x+5|+\frac52arctan(u)

undo your substitution as a final step,
and include a constant of integration,

\rm ln|x^2+2x+5|+\frac52arctan\left(\frac{x+1}{2}\right)+c

Hope that helps!
Lemme know if any steps were too confusing.

You might be interested in
PLS help with 2,4,6 I rlly need help understanding help guide me through
valentina_108 [34]
Try to understand this. Hope this help. Answer for 2 only.

3 0
2 years ago
Read 2 more answers
Please help with this
9966 [12]

Answer:

D

Step-by-step explanation:

The first graph curves down

The second graph is not symmetrical on the y axis

The third graph is much higher than the picture

6 0
2 years ago
Which equation is equivalent to (1/3)x=27x+2?
erma4kov [3.2K]
I hope this helps you


x=3. (27x+2)

x=81x+6

80x=-6

x=-3/40
7 0
2 years ago
Read 2 more answers
17. A population of 20000 rabbits has a growth rate of 25% every three years, Determine the population
Airida [17]

Answer:

abt 36,469

Step-by-step explanation:

36468.75

estimate it to 36,469 bc .75 of rabbit is not possible

3 0
2 years ago
In Chicago, Illinois, the temperature was -14°F in the morning. If the
Rom4ik [11]

Answer:

-20° F

Step-by-step explanation:

the temperature would be :

=》-14 - 6 = -20° F

7 0
2 years ago
Read 2 more answers
Other questions:
  • What is the length of each side of a square that has the same areas as the rectangle below? Round to the nearest tenth. Height:
    12·1 answer
  • Charlie, a science teacher, worked 10 months out of the year. What percent of the year did he work?
    15·2 answers
  • Need help asap! Need to find the value of x
    5·2 answers
  • 3, 5, 9, 15, 23, ___<br> whats the next number
    13·2 answers
  • Petra’s electricity supply company charges her a fixed quarterly sum plus a rate per unit for electricity used. In the most expe
    5·2 answers
  • PLEASE HELP ASAP 25 PTS + BRAINLIEST TO RIGHT/BEST ANSWER
    10·1 answer
  • 3^4 + 2 • 5 = (input whole numbers only.) <br> (please leave explanation)
    11·1 answer
  • The measure is (0.4x + 109)º. Find the value of x
    7·1 answer
  • WHAT VALUE IS EQUIVALENT TO (7+3)^2 + (8-1) X 5 EQUAL
    11·1 answer
  • (7÷3/4)×(16-113/10).​
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!