Mathematics
29.07.2022 01:42
185
248
10
Solved by an expert

(a) Evaluate the integral ∫2048x2+4dx. Your answer should be in the form kπ, where

(a) Evaluate the integral ∫2048x2+4dx. Your answer should be in the form kπ, where k is an integer. What is the value of k? (Hint: darctan(x)dx=1x2+1 ) k= 6 (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x)=48x2+4. Then, integrate it from 0 to 2, and call it S. S should be an infinite series ∑[infinity]n=0an . What are the first few terms of S?
Show Answers
zuleromanos
zuleromanos
4,9(59 marks)

Step-by-step explanation:

a) Evaluate the integral

\int\limits^2_0 {\frac{48}{x^2+4} } \, dx

=\int\limits^2_0 {\frac{48}{x^2+2} } \, dx \\\\=\frac{48}{2} [\tan ^-^1(\frac{\pi}{2}) ]^2_0

=24[\tan ^-^1(1)- \tan^-^1(0)]\\\\=24(\frac{\pi}{4} )\\\\=6\pi\\\\ k \pi=6 \pi\\\\k=6

b)

F(x)=\frac{48}{x^2+4}

divide by 4

=\frac{12}{1+(\frac{x}{2} )^2}

Power series for \frac{1}{1-x}

\frac{1}{1-x} =1+x+x^2+x^3+x^4+...= \sum_{n=0}^{\infty}x^n     |x|

replace x by -\frac{x^2}{4} in equation (1)

\frac{12}{1+(\frac{x}{2})^2 } =12\frac{1}{1-(-\frac{x^2}{4} )}

=12 \sum_{n=0}^{\infty}(-\frac{x^2}{4} )^n\\\\=12\sum_{n=0}^ \infty(-1)^n\frac{x^2^n}{2^2^n}

=12(1-\frac{x^2}{4} +\frac{x^4}{16} -\frac{x^6}{64} +\frac{x^8}{256} -\frac{x^1^0}{1024} ...)

=12-3x^2+\frac{3}{4} x^4-\frac{3}{16} x^6+\frac{3}{64}x^8-\frac{3}{256} x^1^0...

Take integration with respect to x from 0 to 2

\int\limits^2_0 {(12-3x^2+\frac{3}{4}x^4-\frac{3}{16} x^6+\frac{3}{64} x^8-\frac{3}{256} x^1^0+... )} \, dx

=[{12x-x^3+\frac{3}{20} x^5-\frac{3}{16}\frac{x^7}{7} +\frac{3}{64} \frac{x^9}{9} -\frac{3}{256}\frac{x^1^1}{11} +...]^2_0  }

=24-8+4.8-3.42+2.66-2.18\\\\=18.84

mmxotwod
mmxotwod
4,9(81 marks)

How I would write my paragraph:

If angle 1 is congruent to angle 4, angle 1 must also be congruent to angle 3 by the transitive property. This is because is angle 3 and angle 4 are vertical angles, and they must be congruent. If angle 1 is congruent to angle 3, then the two lines in the middle must also be parallel due to the converse of the corresponding angles theorem. Because the two lines in the middle are parallel, angle 3 must be congruent to angle 2 by the alternate interior angles theorem.

Popular Questions about the subject: Mathematics

Find the lateral surface area of the triangular prism. 10 ft...
Mathematics
10.10.2021 11:48
-8x-11x what is the answer...
Mathematics
25.04.2020 19:13
Help please! find the area of the isoceles triangle...
Mathematics
12.02.2020 23:58
WHAT IS 5,350 DIVIDED BY NOTHING =5.35...
Mathematics
15.04.2023 18:44
Find the lateral surface area of the triangular prism : 10ft,15ft,13ft,13ft,13ft,13ft...
Mathematics
26.12.2022 18:04
HELP FAST PLEASE Sketch 0=11pi/6 in standard position...
Mathematics
24.05.2020 15:17
HELP FAST PLEASE sketch 0=5pi/4 in standard position...
Mathematics
08.11.2022 04:27
How many 2 /3 parts are in 2 wholes? The number of 2 /3 parts...
Mathematics
22.06.2020 05:18

New questions by subject

(-7xy/3x)+(4y^2/2y) i also need the steps...
Mathematics
28.07.2020 00:40
Isend letters notes and emai message where does commas goes...
English
22.01.2021 07:22
Is it the responsibility of the united states to spread and...
History
07.07.2022 20:34
Draw a picture to show what 5/4 means. i don t get this...
Mathematics
12.10.2021 20:58
Did the pendleton act go far enough to correct the problems...
History
14.04.2021 09:02
A40 kg skier starts at the top of 12 meter high slope. at the...
Physics
28.04.2022 12:26
Who would benefit from monroe doctrine...
History
09.09.2022 01:38
#
#
#
#
# #

We expand our knowledge with many expert answers