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lisabon 2012 [21]
3 years ago
5

Workout to the simplest:

5E%7B3%7D%20%29%20dx" id="TexFormula1" title=" \int \: {x}^{2} ln( {x}^{3} ) dx" alt=" \int \: {x}^{2} ln( {x}^{3} ) dx" align="absmiddle" class="latex-formula">
​
Mathematics
2 answers:
goblinko [34]3 years ago
7 0

Answer:

\rm \displaystyle \ln(x)   { {x}^{3} } -  \frac{ {x}^{3} }{3}    + \rm C

Step-by-step explanation:

we would like to integrate the following integration

\displaystyle \int  {x}^{2}   \ln( {x}^{3} ) dx

before doing so we can use logarithm exponent rule in order to get rid of the exponent of ln(x³)

\displaystyle \int 3 {x}^{2}   \ln( {x}^{} ) dx

now notice that the integrand is in the mutilation of two different functions thus we can use integration by parts given by

\rm\displaystyle \int u  \cdot \: vdx = u \int vdx -  \int u' \bigg( \int vdx \bigg)dx

where u' can be defined by the differentiation of u

first we need to choose our u and v in that case we'll choose u which comes first in the guideline ILATE which full from is Inverse trig, Logarithm, Algebraic expression, Trigonometry, Exponent.

since Logarithms come first our

\displaystyle u =   \ln(x)  \quad  \text{and}  \quad v =   {3x}^{2}

and u' is \frac{1}{x}

altogether substitute:

\rm \displaystyle \ln(x)  \int  3{x}^{2} dx -  \int  \frac{1}{x}  \left( \int 3 {x}^{2} dx \right)dx

use exponent integration rule to integrate exponent:

\rm \displaystyle \ln(x)  \int  3{x}^{2} dx -  \int  \frac{1}{x}  \left( 3\frac{ {x}^{3} }{3}  \right)dx

once again exponent integration rule:

\rm \displaystyle \ln(x)   3\frac{ {x}^{3} }{3}  -  \int  \frac{1}{x}  \left( 3\frac{ {x}^{3} }{3}  \right)dx

simplify integrand:

\rm \displaystyle \ln(x)   3\frac{ {x}^{3} }{3}  -  \int  \frac{ 3{x}^{3} }{3x} dx

use law of exponent to simplify exponent:

\rm \displaystyle \ln(x)   \frac{ 3{x}^{3} }{3}  -  \int  \frac{ 3\cancel{ {x}^{3}} }{3 \cancel{x}} dx

\rm \displaystyle \ln(x)   \frac{ 3{x}^{3} }{3}  -  \int  \frac{ 3{x}^{3} }{3} dx

use constant integration rule to get rid of constant:

\rm \displaystyle \ln(x)   \frac{3 {x}^{3} }{3}  -  1  \int  {x}^{2}dx

use exponent integration rule:

\rm \displaystyle \ln(x)   \frac{3 {x}^{3} }{3}  -   \frac{ {x}^{3} }{3}

\rm \displaystyle \ln(x)   { {x}^{3} } -  \frac{ {x}^{3} }{3}

and finally we of course have to add the constant of integration:

\rm \displaystyle \ln(x)   { {x}^{3} } -  \frac{ {x}^{3} }{3}    + \rm C

and we are done!

Roman55 [17]3 years ago
4 0

Answer:

Step-by-step explanation:

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The annual salaries of employees in a large company are approximately normally distributed with a mean of $50,000 and a standard
Anna35 [415]

Answer:

a) 30.85% of people earn less than $40,000

b) 37.21% of people earn between $45,000 and $65,000.

c) 15.87% of people earn more than $70,000

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

\mu = 50000, \sigma = 20000

a.What percent of people earn less than $40,000?

This is the pvalue of Z when X = 40000. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{40000 - 50000}{20000}

Z = -0.5

Z = -0.5 has a pvalue of 0.3085.

30.85% of people earn less than $40,000

b.What percent of people earn between $45,000 and $65,000?

This is the pvalue of Z when X = 65000 subtracted by the pvalue of Z when X = 45000. So

X = 65000

Z = \frac{X - \mu}{\sigma}

Z = \frac{65000 - 50000}{20000}

Z = 0.75

Z = 0.75 has a pvalue of 0.7734.

X = 45000

Z = \frac{X - \mu}{\sigma}

Z = \frac{45000 - 50000}{20000}

Z = -0.25

Z = -0.25 has a pvalue of 0.4013.

0.7734 - 0.4013 = 0.3721

37.21% of people earn between $45,000 and $65,000.

c.What percent of people earn more than $70,000?

This is 1 subtracted by the pvalue of Z when X = 70000. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{70000 - 50000}{20000}

Z = 1

Z = 1 has a pvalue of 0.8413.

1 - 0.8413 = 0.1587

15.87% of people earn more than $70,000

3 0
3 years ago
Find the area of this trapezium.
Ronch [10]

Answer:

The area of Trapezium is 21 cm²

Step-by-step explanation:

Divide the trapezium into a Triangle and Rectangle.

The length of the side is given in the figure behind.

So,

The area of Trapezium

= The area of Triangle + The area of Rectangle

= 1/2 × 4 × 3 + 5 × 3 cm

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= 21 cm²

Thus, The area of Trapezium is 21 cm²

<u>-TheUnknownScientist 72</u>

4 0
3 years ago
4x + 3x -7 =49 i need an answer.
Genrish500 [490]

Answer:

<h2>x = 8</h2>

Step-by-step explanation:

4x+3x-7=49\qquad\text{combine like terms}\\\\(4x+3x)-7=49\\\\7x-7=49\qquad\text{add 7 to both sides}\\\\7x-7+7=49+7\\\\7x=56\qquad\text{divide both sides by 7}\\\\\dfrac{7x}{7}=\dfrac{56}{7}\\\\x=8

6 0
3 years ago
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Which of the following figures uses the SSS theorem to prove that AEFG SAHIJ?
jeka94

Answer:

W.

Step-by-step explanation:

SSS, or Side-Side-Side is used to prove two triangles similar

Figure W. is the only one that uses three side marks, all of the others use at least one angle to prove them similar.

5 0
2 years ago
Type the correct answer in each box.
kakasveta [241]

The value of x in A is 7 and the value of y in B is -15

<h3>Functions and values</h3>

Functions are represented as a function of variable. Given the following sets of numbers.

A = [3 x 6 1 -8 5] and;

-5A = [-15 35 -30 -5 40 y]

Equate the individual values to have:

-5A= -15

Divide both sides by -5

-5A/-5 = -15/-5
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Determine the value of y

-5A = y

-5(3) = y

y = -15

Similarly;

5x = 35

x = 35/5

x = 7

Hence the value of x in A is 7 and the value of y in B is -15

Learn more on function and values here: brainly.com/question/10439235

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7 0
1 year ago
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