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
Ber [7]
3 years ago
11

Evaluate the following integral using trigonometric substitution

Mathematics
1 answer:
serg [7]3 years ago
8 0

Answer:

The result of the integral is:

\arcsin{(\frac{x}{3})} + C

Step-by-step explanation:

We are given the following integral:

\int \frac{dx}{\sqrt{9-x^2}}

Trigonometric substitution:

We have the term in the following format: a^2 - x^2, in which a = 3.

In this case, the substitution is given by:

x = a\sin{\theta}

So

dx = a\cos{\theta}d\theta

In this question:

a = 3

x = 3\sin{\theta}

dx = 3\cos{\theta}d\theta

So

\int \frac{3\cos{\theta}d\theta}{\sqrt{9-(3\sin{\theta})^2}} = \int \frac{3\cos{\theta}d\theta}{\sqrt{9 - 9\sin^{2}{\theta}}} = \int \frac{3\cos{\theta}d\theta}{\sqrt{9(1 - \sin^{\theta})}}

We have the following trigonometric identity:

\sin^{2}{\theta} + \cos^{2}{\theta} = 1

So

1 - \sin^{2}{\theta} = \cos^{2}{\theta}

Replacing into the integral:

\int \frac{3\cos{\theta}d\theta}{\sqrt{9(1 - \sin^{2}{\theta})}} = \int{\frac{3\cos{\theta}d\theta}{\sqrt{9\cos^{2}{\theta}}} = \int \frac{3\cos{\theta}d\theta}{3\cos{\theta}} = \int d\theta = \theta + C

Coming back to x:

We have that:

x = 3\sin{\theta}

So

\sin{\theta} = \frac{x}{3}

Applying the arcsine(inverse sine) function to both sides, we get that:

\theta = \arcsin{(\frac{x}{3})}

The result of the integral is:

\arcsin{(\frac{x}{3})} + C

You might be interested in
800 students are going on a school trip by bus.
Tema [17]
The answer is twenty four
6 0
3 years ago
Read 2 more answers
What is greater: 1) 0.48 or 6/15th? 2) 0.75 or 7/8th? 3) 7.025 or 7 1/8ths?
fgiga [73]
1) 0.48 3     2) 7/8     3) 7 1/8
4 0
3 years ago
Arianna gathered data about the distance of 20 cities from the equator and the average hours of sunlight per day in December for
stich3 [128]

<u>Answer-</u>

<em>A. strong negative correlation.</em>

<u>Solution-</u>

<u>Direction of a relationship</u>

  1. Positive- If one variable increases, the other tends to also increase. If one decreases, the other tends to also. It is represented by positive numbers(i.e 0 to 1).
  2. Negative- If one variable increases, the other tends to decrease, and vice-versa.  It is represented by negative numbers(i.e 0 to -1)

<u>Strength of a relationship</u>

  1. Perfect Relationship- When two variables are linearly related, the correlation coefficient is either 1 or -1. They are said to be perfectly linearly related, either positively or negatively.
  2. No relationship- When two variables have no relationship at all, their correlation is 0.

As in this case, correlation coefficient was found to be -0.91, which is negative and close to -1, so it is a strong negative correlation.


5 0
3 years ago
Answer choices 6 cubic units8 cubic units 10.6 cubic units 18 cubic units
antiseptic1488 [7]

Given,

Cylinder A has a volume of 6 cubic units

and height =3 units

The radius of cylinder A,

\begin{gathered} r=\sqrt[]{\frac{V}{\pi h}} \\ =\sqrt[]{\frac{6}{3\pi}} \\ =0.8 \end{gathered}

To find the volume of a cylinder B

\begin{gathered} V=\pi r^2h \\ =(3.14)\times(0.8)^2\times(3) \\ =6.03 \end{gathered}

Thus the volume of cylinder B is 6.03

7 0
1 year ago
What is the approximate area of the circle, in square centimeters? Use 3.14 for π (pi). Record your answer on the grid. then fil
iragen [17]
92 ……0.00.0.0.0.0.00..00.0.0.0. Hours
4 0
3 years ago
Read 2 more answers
Other questions:
  • Kenneth drove 177.55 miles in 3 7/20 hours. what was his average speed
    9·2 answers
  • Find the equation for the line that is parallel to the line with the equation y=4 passing through the point (-4,4).
    8·1 answer
  • Hilary wonders how much sunlight really affects plant growth. She pots 22 plants and puts some in direct sunlight and some in sh
    12·2 answers
  • Identify the error or errors in this argument that supposedly shows that if ∃xp (x) ∧ ∃xq(x) is true then ∃x(p (x) ∧ q(x)) is tr
    14·1 answer
  • I need help on problem 1
    8·1 answer
  • Find the side length of a square whose area is 121 ft^2
    7·2 answers
  • How much does 6 goes into 4,296
    7·2 answers
  • Which equation can be used to find 40 percent of 25? StartFraction 25 divided by 1 Over 40 divided by 1 EndFraction = StartFract
    8·2 answers
  • Solve the inequality -2n &gt; -6<br><br> A. n&gt;3<br> B. n&gt;-3<br> C. n&lt;3<br> D. n=3
    14·1 answer
  • Use the digits 1,2,3,4,5 and the following clues to find the 5-digit number.
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!