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
Helen [10]
2 years ago
13

0%5Ctan%28x%29%29%20%5C%3A%20dx%20%5C%5C%20" id="TexFormula1" title=" \rm \int_{0}^{ \pi } \cos( \cot(x) - \tan(x)) \: dx \\ " alt=" \rm \int_{0}^{ \pi } \cos( \cot(x) - \tan(x)) \: dx \\ " align="absmiddle" class="latex-formula">​
Mathematics
1 answer:
Nikolay [14]2 years ago
7 0

Replace x with π/2 - x to get the equivalent integral

\displaystyle \int_{-\frac\pi2}^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

but the integrand is even, so this is really just

\displaystyle 2 \int_0^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

Substitute x = 1/2 arccot(u/2), which transforms the integral to

\displaystyle 2 \int_{-\infty}^\infty \frac{\cos(u)}{u^2+4} \, du

There are lots of ways to compute this. What I did was to consider the complex contour integral

\displaystyle \int_\gamma \frac{e^{iz}}{z^2+4} \, dz

where γ is a semicircle in the complex plane with its diameter joining (-R, 0) and (R, 0) on the real axis. A bound for the integral over the arc of the circle is estimated to be

\displaystyle \left|\int_{z=Re^{i0}}^{z=Re^{i\pi}} f(z) \, dz\right| \le \frac{\pi R}{|R^2-4|}

which vanishes as R goes to ∞. Then by the residue theorem, we have in the limit

\displaystyle \int_{-\infty}^\infty \frac{\cos(x)}{x^2+4} \, dx = 2\pi i {} \mathrm{Res}\left(\frac{e^{iz}}{z^2+4},z=2i\right) = \frac\pi{2e^2}

and it follows that

\displaystyle \int_0^\pi \cos(\cot(x)-\tan(x)) \, dx = \boxed{\frac\pi{e^2}}

You might be interested in
A triangle has an area of 221.16 in.² the height of the triangle is 29.1 inches what is the length of the base of the triangle
Zigmanuir [339]
The formula you should use is Area = 1/2bh, in other words, area equals half times base times height. Now, substitute everything in, the equation should now be 221.16 = 1/2*b*29.1. This simplifies into 221.16 = 14.55b, now divide 14.55 on both sides, which leaves us to 15.2 = b. However, don't forget the units, which the answer (length of base) should be 15.2 in.^2.
7 0
3 years ago
You have an ant farm with 35 ants. The population of ants in your farm will double every 3 months. The table shows the populatio
vfiekz [6]

Answer:

560 ants and non linear function

Step-by-step explanation:

12 months divided by 3 months is 4. 35 times 2 is 70. Again is 140. Again is 280. Again a 4th time is 560 ants. It is not linear because it does not increase the same amount each time.

7 0
3 years ago
What is the slope of the line through (-1, 2) and (-3,-2)?
Burka [1]

The answer of the slope or Gradient of the line is 2

8 0
2 years ago
Please help please ​
jarptica [38.1K]

Answer:

1.) supplementary  m<1+m<2=180 and m<2+m<3=180

2.) congruent, supplementary

Step-by-step explanation:

the answers are in this order above

5 0
3 years ago
A box of cereal is reduced by 30%. The new box contains 35 ounces. How many ounces did the old box contain?
kumpel [21]

Answer:

45.5

Step-by-step explanation:

6 0
3 years ago
Other questions:
  • Express the number using scientific notation: 57,000,000<br> HELP PLZ!!!
    14·2 answers
  • Taylor is making a cookie cake that has a diameter of 13 in. What is the area of a fourth of the cookie cake?
    5·2 answers
  • bryan is laying down sod in his yard to grow a new lawn.each piede of sod is a 1 foot by 1 foot square. How many pieces of sod w
    8·1 answer
  • What polynomial has a graph that passes through the given points? (-2, 2) (-1, -1) (1, 5) (3, 67) A. y = -x3 + 4x2 + 2x - 2 B. y
    6·2 answers
  • The school choir is selling fruit boxes to earn money for new robes. The fruit boxes sell for $30
    12·1 answer
  • What are the x-intercepts of the graph of the equation above?
    7·2 answers
  • This Question: 1 pt
    10·1 answer
  • A town is going to use its property tax records to pick
    8·1 answer
  • Once upon a time I weighed 120 pounds. Now I weigh 225 pounds. What is the percent change?
    6·2 answers
  • One hundred tomato plants were randomly selected from a large garden, and a 90% confidence interval was computed to determine th
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!