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3 years ago

Much help needed please

Mathematics

2 answers:

alisha [4.7K]3 years ago

6 0

Answer: 0.8

Step-by-step explanation:

Jlenok [28]3 years ago

3 0

There is 9 counters all together and if 2 are picked out at random which are red then it would be 2/9

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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

Yesterday, there were b milligrams of bacteria in a lab experiment. Today, there are b^2 milligrams of bacteria. If there are 40

Ksju [112]

Answer:

Step-by-step explanation:

b^2 = 400

b = 20.............

4 0

3 years ago

On a coordinate plane, a line goes through (0, 4) and (4, 16). The line representing the first table is shown in the graph. What

puteri [66]

Answer:

x=3

y=13

Step-by-step explanation:

okay so what that means is that the two equations are y=-x+16 and y=3x+4

so y+x=16

y-3x=4

so 4x=12 so x=3

y=13 is the solution

4 0

3 years ago

Read 2 more answers

A 25-foot ladder leans against a house. The bottom of the ladder is 7 feet from the house.

Liono4ka [1.6K]

It makes a 74 degree angle.

The sides of the triangle that are given are the side adjacent to the angle the ladder makes with the ground and the hypotenuse of the triangle.

adjacent/hypotenuse is the ratio for cosine:

cos x = 7/25

To solve this, we will take the inverse cosine:

x = cos⁻¹(7/25) = 73.7 = 74

8 0

4 years ago

Number of learners in a class of 45 if x are absent

GalinKa [24]

Answer:

45 - x

Step-by-step explanation:

Number of learners is number in class minus those absent , that is

number of learners = 45 - x

7 0

3 years ago

Much help needed please ​

Much help needed please