A couple of Integration problems

[star001]

A couple of Integration problems 9dx/(x(x^4 + 8 ))

Homework Statement

Hi

I'm new to this forum. I have a couple of Integration Problems which I am not able to integrate correctly. I will also post my attempt at solving the problem so u guys can see what method i took trying to solve these. spent a lot of time in these questions and finally decided to post in this forum.

1
9dx/(x(x^4 + 8 ))

2
integ 37dx/((root(x)+ xroot(x))

The Attempt at a Solution

My attempt

i tried u substitution with u=x^2
also 9 is a constant so i took it out for the time being(ill multiply the answer i get with 9)

9 integ dx/(x(x^4 + 8 )) u = x^2 du=2xdx
9 integ xdx/(x^2(x^4 + 8 ))
9*1/2 integ du/(u(u^2 + 8 )

9*1/2 integ du/(u^3 + 8u ) ??
9*1/2 (ln|u^3 +8u|) <<is that answer correct?

if not can someone kindly tell me how to do this problem pls.




I do not know how to go about the 2nd problem. any hints on how to approach it are welcome :)


thanks in advance!

 

[Donaldos]

1. Decompose \frac{1}{x\left(x^4+8\right)} into partial fractions

2. u=\sqrt{x} + partial fraction decomposition

 

[Mark44]

Homework Statement

Hi

I'm new to this forum. I have a couple of Integration Problems which I am not able to integrate correctly. I will also post my attempt at solving the problem so u guys can see what method i took trying to solve these. spent a lot of time in these questions and finally decided to post in this forum.

1
9dx/(x(x^4 + 8 ))

2
integ 37dx/((root(x)+ xroot(x))

The Attempt at a Solution

My attempt

i tried u substitution with u=x^2
also 9 is a constant so i took it out for the time being(ill multiply the answer i get with 9)

9 integ dx/(x(x^4 + 8 )) u = x^2 du=2xdx
9 integ xdx/(x^2(x^4 + 8 ))
9*1/2 integ du/(u(u^2 + 8 )

9*1/2 integ du/(u^3 + 8u ) ??

So far so good, but the next line is not correct.
\int \frac{du}{u}~=~ln|u| + C
but you don't have just exactly u in the denominator; you have u³ + 8u. To work through that integral you probably need a technique called partial fraction decomposition, AKA partial fractions.

9*1/2 (ln|u^3 +8u|) <<is that answer correct?

if not can someone kindly tell me how to do this problem pls.




I do not know how to go about the 2nd problem. any hints on how to approach it are welcome :)


thanks in advance!

For your second problem, I would start with a substitution u = sqrt(x), and see where that takes you.

 

[star001]

Thanks a lot guys!