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Im having some difficulty with these question, I do kinda know where to start but Im just missing something

 

1. For positive integer n, represent it in terms of n, x, and I(n-1) \(I(n) = \int \log(x)^n \, dx.\)

 

 

 

2. Compute \(\int 4x^3e^{x^2}\,dx \)

 

 

3. Compute \(\int_0^{10} xg''(x)\,dx \)

using this the values g(x): (0,20), (2,10), (4,5), (6,19), (8,17), (10,2)

                                  g'(x): (0,-9), (2,-2), (4,9), (6,10), (8,0), (10,-4)

 Jan 26, 2021
 #1
avatar+112523 
+1

Please just put one question per post.

 

Question 2

 

let     

 \(u=2x^2   \quad and  \quad v'=2xe^{x^2}\)

 

Now  use integration by parts to solve.    

 

Come back here with your answer.

 Jan 27, 2021
 #4
avatar
+1

ok,  i will remeber that for next time.

 

the answer to 2) is 2x^(2)(e^x^2) - (e^x^2) + C

and the answer to 3) is -22

Guest Jan 27, 2021
 #6
avatar+112523 
+1

Good work and thanks for responding.

We always want feedback.

 

I got the same answers as you. 

Although my personal preference is to factorize the Q2 answer.

Melody  Jan 27, 2021
 #2
avatar+112523 
+1

question 3

 

let 

\(u=x \qquad and \qquad v'=g''(x)\)

 

And again use integration by parts

 

 

 

Question 1: 

I do not understand the question. 

 Jan 27, 2021
 #3
avatar+31703 
+3

I suspect question 1 is meant to look like the following (also integration by parts):

 

Alan  Jan 27, 2021
 #5
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I had the idea to turn Log(x)^n into n * log(x), I also noticed you use I(n-2) as a part of the equation which I think is not belonging in the question

Guest Jan 27, 2021
 #7
avatar+186 
+2

Here's a slightly different parts routine

Let

\(\displaystyle I_{n}=\int1.\ln(x)^{n}dx\\ \text{and let } u=\ln(x)^{n}, \quad dv = 1.dx \)

 

Then

\(\displaystyle \frac{du}{dx}=n.\ln(x)^{n-1}\frac{1}{x},\quad v = x\)

so

\(\displaystyle I_{n}=x\ln(x)^{n}-\int xn\ln(x)^{n-1}\frac{1}{x}dx \\ I_{n}=x\ln(x)^n-nI_{n-1}.\)

 Jan 28, 2021
 #8
avatar+31703 
0

Much neater Tiggsy.

Alan  Jan 28, 2021

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