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sesenic [268]
2 years ago
11

Evaluate the integral, show all steps please!

Mathematics
1 answer:
zalisa [80]2 years ago
4 0

Answer:

\dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \text{C}

Step-by-step explanation:

<u>Fundamental Theorem of Calculus</u>

\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x

Take <u>partial fractions</u> of the given fraction by writing out the fraction as an <u>identity</u>:

\begin{aligned}\dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A}{x-4}+\dfrac{B}{x+2}\\\\\implies \dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A(x+2)}{(x-4)(x+2)}+\dfrac{B(x-4)}{(x-4)(x+2)}\\\\\implies x+5 & \equiv A(x+2)+B(x-4)\end{aligned}

Calculate the values of A and B using substitution:

\textsf{when }x=4 \implies 9 = A(6)+B(0) \implies A=\dfrac{3}{2}

\textsf{when }x=-2 \implies 3 = A(0)+B(-6) \implies B=-\dfrac{1}{2}

Substitute the found values of A and B:

\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x = \int \dfrac{3}{2(x-4)}-\dfrac{1}{2(x+2)}\:\:\text{d}x

\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int ax^n\:\text{d}x=a \int x^n \:\text{d}x$\end{minipage}}

If the terms are multiplied by constants, take them outside the integral:

\implies \displaystyle \dfrac{3}{2} \int \dfrac{1}{x-4}- \dfrac{1}{2} \int \dfrac{1}{x+2}\:\:\text{d}x

\boxed{\begin{minipage}{5 cm}\underline{Integrating}\\\\$\displaystyle \int \dfrac{f'(x)}{f(x)}\:\text{d}x=\ln |f(x)| \:\:(+\text{C})$\end{minipage}}

\implies \dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \text{C}

Learn more about integration here:

brainly.com/question/27805589

brainly.com/question/28155016

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3241004551 [841]

Answer:

i think you just extend the coordinates to the side, except the right point, by 3, and then the bottom ones go down by 3, and the top one goes up by 3

Step-by-step explanation:

7 0
3 years ago
1.) What is the average rate of change between x=2 and x=5?
Nimfa-mama [501]

Answer:

Rate of change = 150%

Rate of change = 100%

Rate of change = 100%

Step-by-step explanation:

Find:

Rate of change

x=2 and x=5

x=3 and x=6

x=2 and x=4

Rate of change = [(5-2)/2]100

Rate of change = 150%

Rate of change = [(6-3)/3]100

Rate of change = 100%

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Rate of change = 100%

4 0
3 years ago
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Write as a single power, then<br> evaluate.<br> (3^3)^4
kvasek [131]

Answer:

<h3>531,441</h3>

Step-by-step explanation:

{ ({3}^{3}) }^{4}  \\  =  {3}^{12}  \\  = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \\  = 531441 \\

8 0
3 years ago
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In your random sample of 25 Chips Ahoy bags, you find a mean of 985 chips per bag, with a standard deviation of 10 chips. Run a
stealth61 [152]

Answer:

The p -value is less than 0.001.

Step-by-step explanation:

Given information:

Sample size = 25 chips

Sample mean = 985

Sample standard deviation = 10

Let as assume that sample is distributed normally.

The formula for test statistic

t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}

where,

\overline{x} is sample mean

μ is population mean.

s is sample standard deviation.

n is sample size.

The value of test statistic is

t=\frac{985-1000}{\frac{10}{\sqrt{25}}}

t=\frac{985-1000}{\frac{10}{\sqrt{25}}}

t=-7.5

The p-value is

p=2\times p(t

Therefore the p -value is less than 0.001.

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Answer:

y=65

x=50

Step-by-step explanation:

exterior angle of a triamgle is equal to sum of two interior angles

so

2y=130(as it is isosceles triangle)

y=65

we know y+y+x=180(sum of all angles of triangle)and (vertically opposite angle)

therefore, x=50

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