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3 years ago

Find mZXZY (8x - 1) (5x + 11)

Mathematics

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OLga [1]3 years ago

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Answer:

Can you give options because we unable to answer

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

Step2247 [10]

Answer:

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

Step-by-step explanation:

Fundamental Theorem of Calculus

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

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 integral:

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

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 Integration by Substitution:

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

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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2 years ago

Why is a recipe a good example of procedural writing? Check all that apply.

Dahasolnce [82]

Answer: A, B, D, F

Explanation:

Edge 2022

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2 years ago

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Simplify -|-5 + 2|. this is all of the question

antiseptic1488 [7]

Answer:

-3

Step-by-step explanation:

-(absolute value of -3)

-(3)

-3

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4 years ago

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Estimate the sum of 513.98 and 405.56 to the nearest whole number?

bogdanovich [222]

513.98= 514

405.56=406

dont forget to make me the brainliest

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4 years ago

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Giselle graphs the function f(x) = x2. Robin graphs the function g(x) = –x2. How does Robin’s graph relate to Giselle’s? Robin’s

Sergio039 [100]

Answer: First Option: Robin's graph is a reflection of Giselle´s graph over the x-axis.

A reflection of the graph f(x)=y is g(x)=-y. In this case y=x^2, then:
g(x)=-x^2 that corresponds with the Robin's function

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3 years ago

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