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3 years ago

What is the length of the hypotenuse? O 6 cm O 62 cm O 12 cm O 122 cm

Mathematics

1 answer:

Naya [18.7K]3 years ago

8 0

Is there a picture of the triangle or any measurements other than multiple choice

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Evaluate the integral following

alina1380 [7]

Answer:

$\displaystyle{4\tan x + \sin 2x - 6x + C}$

Step-by-step explanation:

We are given the integral of:

$\displaystyle{\int 4(\sec x - \cos x)^2 \, dx}$

First, we can use a property to separate a constant out of integrand:

$\displaystyle{4 \int (\sec x - \cos x)^2 \, dx}$

Next, expand the expression (integrand):

$\displaystyle{4 \int \sec^2 x - 2\sec x \cos x + \cos^2 x \, dx}$

Since $\displaystyle{\sec x = \dfrac{1}{\cos x}}$ then it can be simplified to:

$\displaystyle{4 \int \dfrac{1}{\cos^2 x} - 2\dfrac{1}{\cos x} \cos x + \cos^2 x \, dx}\\\\\displaystyle{4 \int \dfrac{1}{\cos^2 x} - 2 + \cos^2 x \, dx}$

Recall the formula:

$\displaystyle{\int \dfrac{1}{\cos ^2 x} \, dx = \int \sec ^2 x \, dx = \tan x + C}\\\\\displaystyle{\int A \, dx = Ax + C \ \ \tt{(A \ and \ C \ are \ constant.)}$

For $\displaystyle{\cos ^2 x}$ , we need to convert to another identity since the integrand does not have a default or specific integration formula. We know that:

$\displaystyle{2\cos^2 x -1 = \cos2x}$

We can solve for $\displaystyle{\cos ^2x}$ which is:

$\displaystyle{2\cos^2 x = \cos2x+1}\\\\\displaystyle{\cos^2x = \dfrac{\cos 2x +1}{2}}$

Therefore, we can write new integral as:

$\displaystyle{4 \int \dfrac{1}{\cos^2 x} - 2 + \dfrac{\cos2x +1}{2} \, dx}$

Evaluate each integral, applying the integration formula:

$\displaystyle{\int \dfrac{1}{\cos^2x} \, dx = \boxed{\tan x + C}}\\\\\displaystyle{\int -2 \, dx = \boxed{-2x + C}}\\\\\displaystyle{\int \dfrac{\cos 2x +1}{2} \, dx = \dfrac{1}{2}\int \cos 2x +1 \, dx}\\\\\displaystyle{= \dfrac{1}{2}\left(\dfrac{1}{2}\sin 2x + x\right) + C}\\\\\displaystyle{= \boxed{\dfrac{1}{4}\sin 2x + \dfrac{1}{2}x + C}}$

Then add all these boxed integrated together then we'll get:

$\displaystyle{4\left(\tan x - 2x + \dfrac{1}{4}\sin 2x + \dfrac{1}{2} x\right) + C}$

Expand 4 in the expression:

$\displaystyle{4\tan x - 8x +\sin 2x + 2 x + C}\\\\\displaystyle{4\tan x + \sin 2x - 6x + C}$

Therefore, the answer is:

$\displaystyle{4\tan x + \sin 2x - 6x + C}$

4 0

8 months ago

in a chemistry class, the ratio of girls to boys is 2 to 1. if there are 30 students in the class how many girls are there

VLD [36.1K]

Firstly, we can simplify the ratio of boys to girls to
1
:
2
.
Then, to find out how many students each ratio represents, we add up
1 and 2 to get 3
(1+2=3)
By dividing 3
by the number of students, we can find how many students ONE ratio represents:
24 3=8

So ONE ratio is equal to 8
boys. Since our simplified ratio of boys to girls is
1:2
already, we do not have to do further multiplying - the number of boys is simply 8.
For the girls, simply multiply 2 by 8 to get 16

Check: 8 boys + 18 girls = 24students

5 0

2 years ago

What is the slope of each line?

Paladinen [302]

Try going to desmos.com it helps

8 0

3 years ago

Read 2 more answers

This table shows the number of cars that a local car dealership sold each week for the past five weeks.

xeze [42]

Answer:

it affects the mean because it doesn't affect the mode neither the median so it's the mean

5 0

3 years ago

How does the graph of y = sqrt(x) + 2 compare to the graph of the parent square root function

earnstyle [38]

To move te function up z units, add zto the whole equation

from
y=sqrt(x) to
y=sqrt(x)+2

parenth function moved up 2 units
my hands are cold so I don't type well

8 0

3 years ago

Read 2 more answers