Solve the trigonometric equation for all values 0 ≤ x

Akimi4 [234]

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Evaluate the indefinite integral:

$\mathsf{\displaystyle\int\!\dfrac{1}{(1-y)^2}\,dy}\\\\\\ =\mathsf{\displaystyle\int\!\dfrac{-1}{(1-y)^2}\cdot (-1)\,dy}\\\\\\ =\mathsf{\displaystyle- \int\!(1-y)^{-2}\cdot (-1)\,dy\qquad\quad(i)}$

Substitution:

$\begin{array}{lcl} \mathsf{1-y=u}&\quad\Rightarrow\quad&\mathsf{y=1-u}\\\\ &&\mathsf{dy=-1\,du} \end{array}$

So the integral $\mathsf{(i)}$ becomes

$=\mathsf{\displaystyle -\int\! u^{-2}\,du}\\\\\\ =\mathsf{-\,\dfrac{u^{-2+1}}{-2+1}+C}\\\\\\ =\mathsf{-\,\dfrac{u^{-1}}{-1}+C}\\\\\\ =\mathsf{u^{-1}+C}\\\\\\ =\mathsf{\dfrac{1}{u}+C}$

$=\mathsf{\dfrac{1}{1-y}+C}\\\\\\\ \therefore~~\boxed{\begin{array}{c}\mathsf{\displaystyle\int\!\dfrac{1}{(1-y)^2}\,dy=\dfrac{1}{1-y}+C} \end{array}}\qquad\quad\checkmark$

I hope this helps. =)

Tags: <em>indefinite integral fraction rational substitution power differential calculus</em>

3 0

3 years ago

When solving a system of two equations via substitution, there will always be only one answer.

sertanlavr [38]

No false.
Remember the solution(s) to a system of equations is where the graphs all intersect if at all.
For the case of say a system of 2 lines, you can see the different possible outcomes..
-- If the two lines intersect at some point (x,y), that is one unique solution
-- If the two lines are parallel to each other, you see there are no intersection points and therefore this system of two parallel lines has no solutuion.
-- If the two lines overlap, really the same line written as a multiple of the other line, then you see they intersect at all points along the line, here there are infinite solutions.

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3 years ago

The table shows the cost for ordering a certain number of tacos. What is the value of x if the cost is proportional to the numbe

schepotkina [342]

Answer:

it would be tacos times 1.3 so $7.80

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3 years ago

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You uñere línea the niñez Andes circulé the One but

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What is a fraction equivalent to -6/1 ?

jek_recluse [69]

Answer:

Fractions are equivalent if their simplest forms are the same. So, to make equivalent fractions, all you have to do is multiply the top number (the numerator) and the bottom number (the denominator) by the same number—and it can be any number, as long as you use the same one for both the top and bottom.

Step-by-step explanation:

Fractions are equivalent if their simplest forms are the same. So, to make equivalent fractions, all you have to do is multiply the top number (the numerator) and the bottom number (the denominator) by the same number—and it can be any number, as long as you use the same one for both the top and bottom.

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3 years ago

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Solve the trigonometric equation for all values 0 ≤ x < 2π.