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3 years ago

8

Evaluate the function over the domain {-2, -1, 0, 1, 2, 3}. As the values of the domain increase, do the values of the range inc

rease or decrease?
y= 3.5^x
when x= -2, y=_

1 answer:

coldgirl [10]3 years ago

8 0

Answer:

hhhhhhhjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

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Which two hundred thousands is 185,104 between

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If I understand correctly, I'm guessing you mean which 100,000s(ex.100,000) so it would be between 100,000 and 200,000

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4 years ago

Evaluate the following integral (Calculus 2) Please provide step by step explanation!

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Answer:

$\displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x=-\dfrac{2}{x+1}+\text{C}$

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<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 <u>constant of integration</u>.

<u>Given integral</u>:

$\displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x$

Factor the denominator:

$\begin{aligned}\implies x^2+2x+1 & = x^2+x+x+1\\& = x(x+1)+1(x+1)\\& = (x+1)(x+1)\\& = (x+1)^2\end{aligned}$

$\implies \displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x=\int \dfrac{2}{(x+1)^2}\:\:\text{d}x$

$\textsf{Apply exponent rule} \quad \dfrac{1}{a^n}=a^{-n}$

$\implies \displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x=\int 2(x+1)^{-2}\:\:\text{d}x$

$\boxed{\begin{minipage}{4 cm}\underline{Integrating $ax^n$}\\\\$\displaystyle \int ax^n\:\text{d}x=\dfrac{ax^{n+1}}{n+1}+\text{C}$\end{minipage}}$

Use <u>Integration by Substitution</u>:

$\textsf{Let }u=(x+1) \implies \dfrac{\text{d}u}{\text{d}x}=1 \implies \text{d}x=\text{d}u}$

Therefore:

$\begin{aligned}\displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x & = \int 2(x+1)^{-2}\:\:\text{d}x\\\\& = \int 2u^{-2}\:\:\text{d}u\\\\& = \dfrac{2}{-1}u^{-2+1}+\text{C}\\\\& = -2u^{-1}+\text{C}\\\\& = -\dfrac{2}{u}+\text{C}\\\\& = -\dfrac{2}{x+1}+\text{C}\end{aligned}$

Learn more about integration here:

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