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pantera1 [17]
2 years ago
10

Why is the answer to this integral's denominator have 1+pi^2

Mathematics
1 answer:
ss7ja [257]2 years ago
6 0

It comes from integrating by parts twice. Let

I = \displaystyle \int e^n \sin(\pi n) \, dn

Recall the IBP formula,

\displaystyle \int u \, dv = uv - \int v \, du

Let

u = \sin(\pi n) \implies du = \pi \cos(\pi n) \, dn

dv = e^n \, dn \implies v = e^n

Then

\displaystyle I = e^n \sin(\pi n) - \pi \int e^n \cos(\pi n) \, dn

Apply IBP once more, with

u = \cos(\pi n) \implies du = -\pi \sin(\pi n) \, dn

dv = e^n \, dn \implies v = e^n

Notice that the ∫ v du term contains the original integral, so that

\displaystyle I = e^n \sin(\pi n) - \pi \left(e^n \cos(\pi n) + \pi \int e^n \sin(\pi n) \, dn\right)

\displaystyle I = \left(\sin(\pi n) - \pi \cos(\pi n)\right) e^n - \pi^2 I

\displaystyle (1 + \pi^2) I = \left(\sin(\pi n) - \pi \cos(\pi n)\right) e^n

\implies \displaystyle I = \frac{\left(\sin(\pi n) - \pi \cos(\pi n)\right) e^n}{1+\pi^2} + C

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The test statistic value lies to the right of the critical value. So we have sufficient evidence to reject the null hypothesis.

<h3>What are null hypotheses and alternative hypotheses?</h3>

In null hypotheses, there is no relationship between the two phenomenons under the assumption or it is not associated with the group. And in alternative hypotheses, there is a relationship between the two chosen unknowns.

Students who take a statistics course are given a pre-test on the concepts and skills for the first chapter of a statistics course.

Then they are given a post-test once the professor has concluded lecturing on the material.

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Then we have

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Then the null hypotheses and alternative hypotheses will be

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Hₐ: \mu _d > 0

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Then

\rm S_d = 6.02

The test statistic value is given by

\rm t = \dfrac{\overline{x} _d }{\dfrac{S_d}{\sqrtn}} \\\\t = \dfrac{9.5}{\dfrac{6.02}{\sqrt4}}\\\\t = 3.16

Since this is a right-tailed test, so the critical value is given by

\rm t_{n-1}(\alpha ) = t_3 (0.05) = 2.353

Since the test statistic value lies to the right of the critical value. So we have sufficient evidence to reject the null hypothesis.

Hence, we can conclude that \mu _d > 0 that is test scores have improved.

More about the null hypotheses and alternative hypotheses link is given below.

brainly.com/question/9504281

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7 0
2 years ago
+Brainliest for this one, pts stay 10. Good luck :)
Anon25 [30]

Answer:

7.25 × 10^6

Step-by-step explanation:

1) put into standard form: 6,000,000 + 1,250,000

2) add: 6,000,000 + 1,250,000= 7,250,000

3) put into scientific notation: 7.25 × 10^6

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180 = 180

Answer:

m∠g = 61°

6 0
3 years ago
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