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
Which is greater 8/10 or 63/100 ?
bazaltina [42]
8/10 is greater than 63/100, it is grater because when u change them into a percentage 8/10 is higher
4 0
3 years ago
-19-5/4v is less than or equal too -14
Nesterboy [21]

Answer:

−20.25

Step-by-step explanation:

5 0
2 years ago
Renting video games from Store A costs $3.50 per game plus a monthly fee of $7.50. Renting video games from Store B costs $5.00
Burka [1]

Answer:

A

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Math Question:<br> Pls help I will do anything
kow [346]

Answer:

The probability of spinning red is \frac{1}{3}, the probability of spinning a 1 is \frac{1}{6}, the probability of spinning an odd number is \frac{1}{2}, and the probability of spinning a 9 is 0.

Step-by-step explanation:

Since there are a total of 6 possible outcomes, you can use this in the following questions. There are 2 red tiles, so the probability of spinning red is \frac{2}{6} or \frac{1}{3}. Since there are only one 1 tile, the probability of spinning a 1 is \frac{1}{6}. Since there are three odd numbers, the probability of spinning an odd number is \frac{1}{2}. Since there are no 9 in the spinner, the probability of spinning a 9 is 0 or never.

5 0
2 years ago
If you know that the exact volume of a cylinder is 500 π cm3, how can you find the approximate volume, without knowing the heigh
OLga [1]
The formula for the volume of a cylinder ( V<span>=πr2</span>h) is useable to find the missing height. Then, to find the radius, use the equation, <span>r = square root of (V / (pi x h)). Plug your numbers into the equation and solve.</span>
5 0
3 years ago
Other questions:
  • 596.282 ×_____ = 5,962.82 <br> A) 1 <br> B) 10 <br> C) 100 <br> D) 1,000
    15·2 answers
  • Why is it that the orthocenter of an obtuse triangle must lie on the outside of the triangle?
    11·1 answer
  • A new patio will be an irregular hexagon. The patio will have two long parallel sides and an area of 360 square feet. The area o
    13·2 answers
  • If she is going x amount of miles an hour how many is she going in 90 minutes
    8·1 answer
  • What is one thousand less than 921,438​
    7·2 answers
  • How many $3.50 can you buy for a million, 10^6, dollars?
    13·1 answer
  • A point
    15·1 answer
  • A rider sits on a motorcycle. The motorcycle has a mass of 237kg. The rider has a mass of 89kg. What is the total mass of the mo
    5·1 answer
  • hii how are yall doing me im ok ig.... i.dk anymore also plzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz help ty
    14·1 answer
  • Which item below has the lowest cost per piece of pizza?
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!