What is the value of Y?

Mathematics

1 answer:

lord [1]2 years ago

8 0

92° i believe im bad at math lol

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∫(cosx) / (sin²x) dx

kirza4 [7]

If you're using the app, try seeing this answer through your browser: brainly.com/question/2822772

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Evaluate the indefinite integral:

$\mathsf{\displaystyle\int\! \frac{cos\,x}{sin^2\,x}\,dx}\\\\\\
=\mathsf{\displaystyle\int\! \frac{1}{(sin\,x)^2}\cdot cos\,x\,dx\qquad\quad(i)}$

Make the following substitution:

$\mathsf{sin\,x=u\quad\Rightarrow\quad cos\,x\,dx=du}$

and then, the integral (i) becomes

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

Integrate it by applying the power rule:

$\mathsf{=\dfrac{u^{-2+1}}{-2+1}+C}\\\\\\
\mathsf{=\dfrac{u^{-1}}{-1}+C}\\\\\\
\mathsf{=-\,\dfrac{1}{u}+C}$

Now, substitute back for u = sin x, so the result is given in terms of x:

$\mathsf{=-\,\dfrac{1}{sin\,x}+C}\\\\\\
\mathsf{=-\,csc\,x+C}$

$\therefore~~\boxed{\begin{array}{c}\mathsf{\displaystyle\int\! \frac{cos\,x}{sin^2\,x}\,dx=-\,csc\,x+C} \end{array}}\qquad\quad\checkmark$

I hope this helps. =)

Tags: <em>indefinite integral substitution trigonometric trig function sine cosine cosecant sin cos csc differential integral calculus</em>

5 0

3 years ago

Wewaii [24]

Answer:

For covering $1$ unit area of the entire playground, the amount of sand required is equal to volume of $3$ buckets of sand.

Step-by-step explanation:

Given -

$\frac{1}{3}$ volume of sand in bucket is able to cover $\frac{1}{9}$ area of the entire playground

Thus,

For covering $\frac{1}{9}$ unit area of the entire playground, the amount of sand required is equal to $\frac{1}{3}$ of the total volume of sand in bucket

For covering $1$ unit area of the entire playground, the amount of sand required is equal to

$\frac{\frac{1}{3} }{\frac{1}{9} } \\\\\frac{1}{3} * \frac{9}{1} \\\frac{9}{3}\\= 3$