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3 years ago

7y + 8y can be simplified to 15y true or false

Mathematics

2 answers:

MatroZZZ [7]3 years ago

8 0

Answer:

True

Step-by-step explanation:

7y and 8y are like terms, so you can add them together

RoseWind [281]3 years ago

7 0

True.

How? By combining like terms.

7y + 8y = what?

We keep the variable y and simply add the numbers.

7 + 8 = 15

We now include y in our answer.

So, 7y + 8y = 15y.

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murzikaleks [220]

Let $x = \arcsin(y)$ , so that

$\sin(x) = y$

$\tan(x)=\dfrac y{\sqrt{1-y^2}}$

$dx = \dfrac{dy}{\sqrt{1-y^2}}$

Then the integral transforms to

$\displaystyle \int_{x=0}^{x=\frac\pi2} \tan(x) \ln(\sin(x)) \, dx = \int_{y=\sin(0)}^{y=\sin\left(\frac\pi2\right)} \frac{y}{\sqrt{1-y^2}} \ln(y) \frac{dy}{\sqrt{1-y^2}}$

$\displaystyle \int_{x=0}^{x=\frac\pi2} \tan(x) \ln(\sin(x)) \, dx = \int_0^1 \frac{y}{1-y^2} \ln(y) \, dy$

Integrate by parts, taking

$u = \ln(y) \implies du = \dfrac{dy}y$

$dv = \dfrac{y}{1-y^2} \, dy \implies v = -\dfrac12 \ln|1-y^2|$

For 0 < y < 1, we have |1 - y²| = 1 - y², so

$\displaystyle \int_0^1 \frac{y}{1-y^2} \ln(y) \, dy = uv \bigg|_{y\to0^+}^{y\to1^-} + \frac12 \int_0^1 \frac{\ln(1-y^2)}{y} \, dy$

It's easy to show that uv approaches 0 as y approaches either 0 or 1, so we just have

$\displaystyle \int_0^1 \frac{y}{1-y^2} \ln(y) \, dy = \frac12 \int_0^1 \frac{\ln(1-y^2)}{y} \, dy$

Recall the Taylor series for ln(1 + y),

$\displaystyle \ln(1+y) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}n y^n$

Replacing y with -y² gives the Taylor series

$\displaystyle \ln(1-y^2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}n (-y^2)^n = - \sum_{n=1}^\infty \frac1n y^{2n}$

and replacing ln(1 - y²) in the integral with its series representation gives

$\displaystyle -\frac12 \int_0^1 \frac1y \sum_{n=1}^\infty \frac{y^{2n}}n \, dy = -\frac12 \int_0^1 \sum_{n=1}^\infty \frac{y^{2n-1}}n \, dy$

Interchanging the integral and sum (see Fubini's theorem) gives

$\displaystyle -\frac12 \sum_{n=1}^\infty \frac1n \int_0^1 y^{2n-1} \, dy$

Compute the integral:

$\displaystyle -\frac12 \sum_{n=1}^\infty \frac1n \int_0^1 y^{2n-1} \, dy = -\frac12 \sum_{n=1}^\infty \frac{y^{2n}}{2n^2} \bigg|_0^1 = -\frac14 \sum_{n=1}^\infty \frac1{n^2}$

and we recognize the famous sum (see Basel's problem),

$\displaystyle \sum_{n=1}^\infty \frac1{n^2} = \frac{\pi^2}6$

So, the value of our integral is

$\displaystyle \int_0^{\frac\pi2} \tan(x) \ln(\sin(x)) \, dx = \boxed{-\frac{\pi^2}{24}}$

6 0

3 years ago

What is the midpoint coordinates of segment HX H(13,8) X(-6,-6)

Setler79 [48]

Midpoint of (x1,y1) and (x2,y2) is

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

so the midpoint of HX is the mdpoint of (13,8) and (-6,-6)
which is

$(\frac{13-6}{2},\frac{8-6}{2}=$
$(\frac{7}{2},\frac{2}{2}=$
$(3.5,1)$

the midpoint is (3.5,1)

7 0

4 years ago

What is 0.70 simplest form

Arturiano [62]

7/10 because there is two places after the decimal point making it 100 so move the decimal point two places to the right which makes it 70 and 70/100 in simplest form is 7/10 <span />

3 0

4 years ago

Which of the following correctly expresses the number below in scientific notation?

Dafna11 [192]

The answer is E. $7.09*10^5$

6 0

3 years ago

PLEASE HELP ASAP AND ONLY IF YOU KNOW THE CORRECT ANSWER<br> Find the missing angle measures.

Gala2k [10]

Answer:

100

Step-by-step explanation:

triangles are equal to 180 degrees so in the one triangle 180-50-50=80 so that missing angel is 80 and then to creat a straight line between the missing angle m<1 and the 80 you subtract the two 180-80=100 leaving the missing angle to be 100

8 0

3 years ago