Answers
The expression gotten from integrating [tex]\int\limits {\frac{1}{\sqrt{100 - 256x\²}} \, dx[/tex] is [tex]\frac{1}{16}\sin^{-1}(8x/5) + c[/tex]
How to integrate the expressionFrom the question, we have the following trigonometry function that can be used in our computation:
[tex]\int\limits {\frac{1}{\sqrt{100 - 256x\²}} \, dx[/tex]
Let u = 8x/5
So, we have
du = 8/5 dx
Subsitute u = 8x/5 and du = 8/5 dx
So, we have
[tex]\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx = \int {\frac{5}{\sqrt{5(100 - 100u\²)}} \, du[/tex]
Simplify
So, we have
[tex]\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{1}{16} \int {\frac{1}{\sqrt{1 -u\²}} \, du[/tex]
Next, we integrate the expression [tex]\int {\frac{1}{\sqrt{1 -u\²}} \, du = \arcsin(u)[/tex]
So, we have
[tex]\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{\arcsin(u)}{16} + c[/tex]
Undo the earlier substitution for u
So, we have
[tex]\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{\arcsin(8x/5)}{16} + c[/tex]
This can also be expressed as
[tex]\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{1}{16}\sin^{-1}(8x/5) + c[/tex]
Hence, integrating the expression [tex]\int\limits {\frac{1}{\sqrt{100 - 256x\²}} \, dx[/tex] gives (c)
[tex]\frac{1}{16}\sin^{-1}(8x/5) + c[/tex]
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Related Questions
Given f(x) = -x² - 10x - 9, find f(-1)
Answers
Answer:
0
Step-by-step explanation:
Given
f ( x ) = - x² - 10x - 9
To find : f ( - 1 )
x = 1
f ( 1 ) = - ( - 1 )² - 10 ( - 1 ) - 9
= - 1 + 10 - 9
= 10 - 9 - 1
= 10 - 10
f ( 1 ) = 0
A toy manufacturer develops a formula to determine the demand for its product depending on the price in dollars. The formula is , where P is the price per unit and D is the number of units in demand. At what price will the demand drop to 584 units?
Answers
The price at which the demand drops to 584 units is $20.80 per unit.
The formula given is:
D = 1000 - 20P
To find the price at which the demand drops to 584 units, we can set D equal to 584 and solve for P:
584 = 1000 - 20P
20P = 1000 - 584
20P = 416
P = 416/20
P = 20.8
Therefore, the price at which the demand drops to 584 units is $20.80 per unit.
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A restaurant charged one customer $28.20 for 3 small servings and 5 large servings. It charged another customer $23.30 for 4 small servings and 3 large servings. How much does one small serving cost?
Answers
Given statement solution is :-One small serving costs approximately $2.90.
Let's assume the cost of one small serving is represented by 'x' dollars.
According to the given information:
Customer 1 received 3 small servings and 5 large servings, for a total cost of $28.20.
Customer 2 received 4 small servings and 3 large servings, for a total cost of $23.30.
We can set up two equations based on the information provided:
Equation 1: 3x + 5y = 28.20 (where 'y' represents the cost of one large serving)
Equation 2: 4x + 3y = 23.30
To solve these equations, we can use the method of substitution or elimination. I'll use the method of substitution:
From Equation 1, we can express y in terms of x:
5y = 28.20 - 3x
y = (28.20 - 3x) / 5
Now we substitute this value of y into Equation 2:
4x + 3((28.20 - 3x) / 5) = 23.30
Multiplying through by 5 to eliminate the denominator:
20x + 3(28.20 - 3x) = 116.50
20x + 84.60 - 9x = 116.50
11x = 116.50 - 84.60
11x = 31.90
x = 31.90 / 11
x ≈ 2.90
Therefore, one small serving costs approximately $2.90.
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