(q48) Solve the integral

Answers

Answer 1

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 expression

From 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]

Related Questions

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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Given f(x) = -x² - 10x - 9, find f(-1)

Answers

f(-1) = -1 + 10 - 9 = 0

Answer:

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

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