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
ANEK [815]
3 years ago
11

SOLUTION We observe that f '(x) = -1 / (1 + x2) and find the required series by integrating the power series for -1 / (1 + x2).

Mathematics
1 answer:
Ann [662]3 years ago
7 0

Answer:

Required series is:

\int{\frac{-1}{1+x^{2}} \, dx =-x+\frac{x^{3}}{3}-\frac{x^{5}}{5}+\frac{x^{7}}{7}+.....

Step-by-step explanation:

Given that

                           f'(x) = -\frac{1}{1 + x^{2}} ---(1)

We know that:

                  \frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^{2}} ---(2)

Comparing (1) and (2)

                           f'(x)=-(tan^{-1}x) ---- (3)

Using power series expansion for tan^{-1}x

f'(x)=-tan^{-1}x=-\int {\frac{1}{1+x^{2}} \, dx

= -\int{ \sum\limits^{ \infty}_{n=0} (-1)^{n}x^{2n}} \, dx

= -\sum{ \int\limits^{ \infty}_{n=0} (-1)^{n}x^{2n}} \, dx

=-[c+\sum\limits^{ \infty}_{n=0} (-1)^{n}\frac{x^{2n+1}}{2n+1}]

=C+\sum\limits^{ \infty}_{n=0} (-1)^{n+1}\frac{x^{2n+1}}{2n+1}

=C-x+\frac{x^{3}}{3}-\frac{x^{5}}{5}+\frac{x^{7}}{7}+.....

as

                 tan^{-1}(0)=0 \implies C=0

Hence,

\int{\frac{-1}{1+x^{2}} \, dx =-x+\frac{x^{3}}{3}-\frac{x^{5}}{5}+\frac{x^{7}}{7}+.....

You might be interested in
A number x is greater than 0. write the word sentence as an inequality
IgorLugansk [536]

9514 1404 393

Answer:

  x > 0

Step-by-step explanation:

The only thing that needs translation to symbols is "greater than". Your "word sentence" already uses the symbols x and 0. The symbol for greater than is >. Your inequality is ...

  x > 0

8 0
3 years ago
Evaluate this expression using the given values for x and y.
nataly862011 [7]

Answer:

The answer is 33.

Step-by-step explanation:

4x^2 + 13 - 2y, x=3 y=8

In order to evaluate this, you just need to plug the x and y in the expression:

= 4(3)^2 + 13 - 2(8)

= 4(9) + 13 - 16

= 36 + 13 - 16

= 33

5 0
3 years ago
Evaluate 0 × |-6|. ......
kupik [55]

Answer:

0

Step-by-step explanation:

Any value multiplied by zero equals zero , thus

0 × | - 6 | = 0

6 0
3 years ago
Which rate converts to a unit rate of 65 miles per hour
bulgar [2K]
(lol were both doing unit rate rn im just pausing to do some brainly questions)

We know that Unit Rate can be found by dividing. 
i think you want the answer right now so if you need an explanation then tell me in the comments
Its number A. Hope this helped!
3 0
3 years ago
Please Help, This is confusing. What is the equation of the line that passes through the point (2,-6) and has a slope of -1?
vovangra [49]

Answer:

ok ok it is y+6=-1(x-2)

Step-by-step explanation:

brainliest?

8 0
3 years ago
Read 2 more answers
Other questions:
  • What is the measure of angle DEF to the nearest degree?
    6·1 answer
  • PLZZZZZZ HLPPPPP MEEEEEEEE what is the factorization of the polynomial graphed below? assume it has no constant factor. write fa
    15·1 answer
  • Write 0.0501 in scientific notation
    15·2 answers
  • Can y’all please help me
    14·1 answer
  • Classify the pair of angles as adjacent, vertical, supplementary, complementary, or neither.
    15·1 answer
  • What property involves using opposites?
    11·2 answers
  • NEED HELP PLEASE AND THANKS!
    10·1 answer
  • What is the perimeter of this rectangle pls help
    12·2 answers
  • Help me solve this pls
    6·1 answer
  • 31 cars are in a race. In how many different ways can cars finish in order in the top 4 positions?
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!