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Alinara [238K]
3 years ago
15

Evaluate using integration by parts ​

Mathematics
1 answer:
PolarNik [594]3 years ago
8 0

Rather than carrying out IBP several times, let's establish a more general result. Let

I(n)=\displaystyle\int x^ne^x\,\mathrm dx

One round of IBP, setting

u=x^n\implies\mathrm du=nx^{n-1}\,\mathrm dx

\mathrm dv=e^x\,\mathrm dx\implies v=e^x

gives

\displaystyle I(n)=x^ne^x-n\int x^{n-1}e^x\,\mathrm dx

I(n)=x^ne^x-nI(n-1)

This is called a power-reduction formula. We could try solving for I(n) explicitly, but no need. n=5 is small enough to just expand I(5) as much as we need to.

I(5)=x^5e^x-5I(4)

I(5)=x^5e^x-5(x^4e^x-4I(3))=(x-5)x^4e^x+20I(3)

I(5)=(x-5)x^4e^x+20(x^3e^x-3I(2))=(x^2-5x+20)x^3e^x-60I(2)

I(5)=(x^2-5x+20)x^3e^x-60(x^2e^x-2I(1))=(x^3-5x^2+20x-60)x^2e^x+120I(1)

I(5)=(x^3-5x^2+20x-60)x^2e^x+120(xe^x-I(0))

Finally,

I(0)=\displaystyle\int e^x\,\mathrm dx=e^x+C

so we end up with

I(5)=(x^4-5x^3+20x^2-60x+120)xe^x-120e^x+C

I(5)=(x^5-5x^4+20x^3-60x^2+120x-120)e^x+C

and the antiderivative is

\displaystyle\int2x^5e^x\,\mathrm dx=(2x^5-10x^4+40x^3-120x^2+240x-240)e^x+C

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J² - jk +k²= 7<br>J⁴ +j²k² + k⁴ = 133​
LekaFEV [45]

Answer: You can search it up...

Step-by-step explanation: Have a nice day!

4 0
3 years ago
For the vectors u and v with magnitudes u = 88 and v = 99​, find the angle θ between u and v which makes u and v = 77.
WARRIOR [948]

Answer:

<h2>89.5°</h2>

Step-by-step explanation:

Using the vector formula u.v = |u||v|cos\theta

|u| = magnitude of vector u

|v| = magnitude of vector v

u.v is the dot product of vector u and v

Given |u| = 88, |v| = 99 and u.v = 77, to get \theta we will substitute the given values into the equation above;

77 = 88*99cos\theta\\cos\theta = \frac{77}{88*99} \\cos\theta = \frac{77}{8712} \\cos\theta = 0.008838\\\theta = cos^{-1} 0.008838\\\theta = 89.5^{0}

4 0
3 years ago
Find the 95% confidence interval for estimating the population mean μ
AVprozaik [17]

We first need to determine whether we are dealing with means or proportions in this problem. Since we are given the sample and population mean, we know that we are dealing with means.

Since we have one sample mean, this means we are creating a confidence interval for one sample (1 Samp T Int).

Normally we would check for conditions, but since this is not formulated as a "real-world scenario" type problem, it is hard to check for randomness and independence. Therefore, I will be excluding conditions from this answer.

<h3>Confidence Interval Formula</h3>

The formula for constructing a <u>confidence interval for means</u> is as follows:

  • \displaystyle \overline{x} \pm t^*\big{(}\frac{\sigma}{\sqrt{n} } \big{)}

We are given these variables:

  • \overline{x}=50
  • n=60
  • \sigma=10

Plug these values into the formula for the confidence interval:

  • \displaystyle 50\pm t^* \big{(}\frac{10}{\sqrt{60} } \big{)}

<h3>Finding the Critical Value (t*)</h3>

In order to find t*, we can use this formula:

  • \displaystyle \frac{1-C}{2}=A

Calculating the z-score associated with "A" will give us t*.

So, let's plug in the confidence interval 95% (.95) into the formula:

  • \displaystyle \frac{1-.95}{2}=.025

Use your calculator or a t-table to find the z-score associated with this area under the curve.. you should get:

  • t^*=1.96

<h3>Constructing Confidence Interval</h3>

Now, let's finish the confidence interval we created:

  • \displaystyle 50\pm 1.96 \big{(}\frac{10}{\sqrt{60} } \big{)}

We can calculate the confidence interval, using this formula, to be:

  • \boxed{(47.4697, \ 52.5303)}

<h3>Interpreting the Confidence Interval</h3>

We are 95% confident that the true population mean μ lies between <u>47.4697 and 52.5303</u>.

8 0
1 year ago
A 12 ft ladder is leaning against a wall with an angle of elevation of 31.2 degrees. How high up the wall will the ladder reach?
Tom [10]

Answer:

12/sin90'=b/sin67'

12sin67'=bsin90'

b=12sin67/sin90'

Step-by-step explanation:

sister ,we can use the law of sines. since we have the angle of elevation and assume the wall makes a right  angle with the ground, our angle opposite the ground is 180-[90+23=67

hope this is  helpful for you

8 0
2 years ago
Help pls and thank you
netineya [11]
A, C, and D are all correct answers, since you are taking away 4 in each of the answers. Hope this helped!
6 0
3 years ago
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