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charle [14.2K]
2 years ago
15

Help me with differentation and integration please!!

Mathematics
1 answer:
Marina86 [1]2 years ago
6 0

Answer:

See below

Step-by-step explanation:

\dfrac{d}{dx} (\tan^3 x) = 3\sec^4 x - 3\sec^2 x

Recall

\dfrac{d}{dx}\tan x=\sec^2

Using the chain rule

\dfrac{dy}{dx}= \dfrac{dy}{du} \dfrac{du}{dx}

such that u = \tan x

we can get a general formulation for

y = \tan^n x

Considering the power rule

\boxed{\dfrac{d}{dx} x^n = nx^{n-1}}

we have

\dfrac{dy}{dx} =n u^{n-1} \sec^2 x \implies \dfrac{dy}{dx} =n \tan^{n-1} \sec^2 x

therefore,

\dfrac{d}{dx}\tan^3 x=3\tan^2x \sec^2x

Now, once

\sec^2 x - 1= \tan^2x

we have

3\tan^2x \sec^2x =  3(\sec^2 x - 1) \sec^2x = 3\sec^4x-3\sec^2x

Hence, we showed

\dfrac{d}{dx} (\tan^3 x) = 3\sec^4 x - 3\sec^2 x

================================================

For the integration,

$\int \sec^4 x\, dx $

considering the previous part, we will use the identity

\boxed{\sec^2 x - 1= \tan^2x}

thus

$\int\sec^4x\,dx=\int \sec^2 x(\tan^2x+1)\,dx = \int \sec^2 x \tan^2x+\sec^2 x\,dx$

and

$\int \sec^2 x \tan^2x+\sec^2 x\,dx = \int \sec^2 x \tan^2x\,dx + \int \sec^2 x\,dx $

Considering u = \tan x

and then du=\sec^2x\ dx

we have

$\int u^2 \, du = \dfrac{u^3}{3}+C$

Therefore,

$\int \sec^2 x \tan^2x\,dx + \int \sec^2 x\,dx = \dfrac{\tan^3 x}{3}+\tan x + C$

$\boxed{\int \sec^4 x\, dx  = \dfrac{\tan^3 x}{3}+\tan x + C }$

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It is estimated that 88% of senior citizens suffer from sleep disorders and 7% suffer from anxiety. Moreover, 5% of senior citiz
Alla [95]

Answer:

a) 0.71

b) 0.06

Step-by-step explanation:

We solve using Baye's Theorem

It is estimated that 88% of senior citizens suffer from sleep disorders and 7% suffer from anxiety. Moreover, 5% of senior citizens suffer from both sleep disorders and anxiety.

We have Two events

A and B

Events A = 88% of senior citizens suffer from sleep disorders

P(A) = 0.88

Event B = 7% suffer from anxiety

P(B) = 0.07

Moreover, 5% of senior citizens suffer from both sleep disorders and anxiety.

P(A and B) = 0.05

a)Given that a senior citizen suffers from anxiety, what is the probability that he or she also suffers from a sleep disorder? Round your answer to the nearest hundredth.

This is calculated as:

P(A and B)/P(B)

= 0.05/0.07

= 0.7142857143

Approximately = 0.71

B) Find the probability that a senior citizen suffers from anxiety, given that he or she has a sleep disorder. Round your answer to the nearest hundredth.

This is calculated as:

P(A and B)/P(A)

= 0.05/0.88

= 0.0568181818

Approximately = 0.06

7 0
2 years ago
Question 6 of 20 :
drek231 [11]

Answer:

The answer is A. (an even number)

Step-by-step explanation:

2 x 27 is equivalent to 54 and 54 is an even number.

7 0
3 years ago
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Please please help me with this and say the answer please asap
lesya692 [45]

Answer:

C, 7 times 7 = 49

7 times 4= 28

7 0
3 years ago
Helpppp pls. Rotate traingle TUV 90 degrees clockwise around the origin
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