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Leya [2.2K]
2 years ago
13

Find the integral ∫√(9+x)/(9-x)

Mathematics
1 answer:
densk [106]2 years ago
6 0

I suppose you mean

\displaystyle \int \frac{\sqrt{9+x}}{9-x} \, dx

Substitute y = √(9 + x). Solving for x gives x = y² - 9, so that 9 - x = 18 - y², and we have differential dx = 2y dy. Replacing everything in the integral gives

\displaystyle \int \frac{2y^2}{18 - y^2} \, dy

Simplify the integrand by dividing:

\dfrac{2y^2}{18 - y^2} = -2 + \dfrac{36}{18 - y^2}

\implies \displaystyle \int \left(\frac{36}{18-y^2} - 2\right) \, dy

For the first term of this new integral, we have the partial fraction expansion

\dfrac1{18 - y^2} = \dfrac1{\sqrt{72}} \left(\dfrac1{\sqrt{18}-y} + \dfrac1{\sqrt{18}+y}\right)

\implies \displaystyle \frac{36}{\sqrt{72}} \int \left(\frac1{\sqrt{18}-y} + \frac1{\sqrt{18}+y}\right) \, dy - 2 \int dy

The rest is trivial:

\displaystyle \sqrt{18} \int \left(\frac1{\sqrt{18}-y} + \frac1{\sqrt{18}+y}\right) \, dy - 2 \int dy

= \displaystyle \sqrt{18} \left(\ln\left|\sqrt{18}+y\right| - \ln\left|\sqrt{18}-y\right|\right) - 2y + C

= \displaystyle \sqrt{18} \ln\left|\frac{\sqrt{18}+y}{\sqrt{18}-y}\right| - 2y + C

= \boxed{\displaystyle \sqrt{18} \ln\left|\frac{\sqrt{18}+\sqrt{9+x}}{\sqrt{18}-\sqrt{9+x}}\right| - 2\sqrt{9+x} + C}

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What number is halfway between 0.2 and 0.3​
MrRissso [65]

Answer:

For numbers between 0.1 and 0.2, there is a 1 in the tenths' place: 0.11, 0.12, 0.13, and so on. For numbers between 0.2 and 0.3, there is a 2 in the tenths' place: 0.21, 0.22, 0.23, and so forth. For example, the number “seven and forty-two hundredths” can be written as 7.42.

Step-by-step explanation:

3 0
3 years ago
Can y’all plz help and thank you. Aslo hope y’all have a wonderful and beautiful day.
stepan [7]

Answer:

a) Rumble: $1

Pedal: $0.6

b) more than 5 songs

c) No, should be integers only

d) No, has to be integers greater than or equal to 2

Step-by-step explanation:

Part a

Rate per song is the slope.

Rumble:

Slope is 0.6

Rate is $0.6 per song

Pedal:

Slope: (4-2)/(2-0) = 1

Rate is $1 per song

Part b

Pedal would be a better deal if it's cost is lower than Rumble's

Rumble's cost: C = 2 + S

Pedal's cost: C = 4 + 0.6S

4 + 0.6S < 2 + S

0.4S > 2

S > 5

c) No, because no. of songs has to be positive integers only.

For example graph shows S = 2.5, but we know that's not possible

d) No, since the cost is 2 + S,

cost can't be in decimals either.

Cost too should be only integers greater than equal to 2

7 0
3 years ago
You ask 200 randomly chosen students to name their favorite ice cream flavor. There are 1800 students in the school. Based on th
Kaylis [27]

Answer:

351

Step-by-step explanation:

Complete question :

Vanilla= 36

Chocolate=21

Strawberry=24

Chocolate Chip=39

SOLUTION:

39 chocolate chip to 200 students = x chocolate chip to 1800 students

39/200 = x / 1800

Solving for 'x'

(200)(x) = (39)(1800)

200x = 70200

x = 70200/200

x = 351

OR

39 out of 200 picked chocolate chip

therefore, 39/200 = 0.195 = 19.5%  ( 32.5% chose chocolate chip)

19.5% of 1800 = 0.195(1800) = 351 students

6 0
3 years ago
Solve the equation.<br><br>-4 (y - 2) = 12<br><br>y = ?​
inna [77]
First you distribute what’s outside the parentheses (the -4)

-4•4= -4y
-4•-2= 8

New equation: -4y+8=12

Now you solve like a normal equation
12-8 is 4

New equation: -4y=4

Now divide

Answer:-1
8 0
3 years ago
For a field trip 20 students rode in cars and the rest filled 5 busses. How many students were in each bus if 210 students were
xeze [42]
First do 210 - 20 = 190
Then do 190/5 = 38
38 people per bus is the answer!!!!
3 0
3 years ago
Read 2 more answers
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