Integration by Partial Fractions Help

[StrangeCharm]

Homework Statement

∫ [x^(3)+4] / [x^(2)+4] dx

Homework Equations

N/A

The Attempt at a Solution

I know that the fraction is improper, so I used long division to rewrite it as x+(-4x+4)/[x^(2)+4].
Given the form S(x)+R(x)/Q(x), Q(x) is a distinct irreducible quadratic factor [x^(2)+4].
I used the rule ax^2+bx+c ⇒ (Ax+B)/(ax^2+bx+c) to rewrite it as (Ax+B)/[x^(2)+4].
I then solved for A and B and got A=-4 and B=4.
I am now trying to solve ∫ [ x + (-4x+4)/(x^(2)+4) ] dx
I know that ∫x=(1/2)x^2, but I am stuck with integrating (-4x+4)/[x^(2)+4].
(I tried u-substitution and that didn't work.)

 

[SammyS]

Homework Statement

∫ [x^(3)+4] / [x^(2)+4] dx

Homework Equations

N/A

The Attempt at a Solution

I know that the fraction is improper, so I used long division to rewrite it as x+(-4x+4)/[x^(2)+4].
Given the form S(x)+R(x)/Q(x), Q(x) is a distinct irreducible quadratic factor [x^(2)+4].
I used the rule ax^2+bx+c ⇒ (Ax+B)/(ax^2+bx+c) to rewrite it as (Ax+B)/[x^(2)+4].
I then solved for A and B and got A=-4 and B=4.
I am now trying to solve ∫ [ x + (-4x+4)/(x^(2)+4) ] dx
I know that ∫x=(1/2)x^2, but I am stuck with integrating (-4x+4)/[x^(2)+4].
(I tried u-substitution and that didn't work.)

Hello StrangeCharm. Welcome to PF !

Split ##\displaystyle \frac{-4x+4}{x^2+4}## into two fractions:

##\displaystyle \frac{-4x}{x^2+4} + \frac{4}{x^2+4}##

The integral of one of them can be done via u-substitution. The other is a fairly well known integral.

 

[StrangeCharm]

Split ##\displaystyle \frac{-4x+4}{x^2+4}## into two fractions:

##\displaystyle \frac{-4x}{x^2+4} + \frac{4}{x^2+4}##

The integral of one of them can be done via u-substitution. The other is a fairly well known integral.

Thanks for the tip!

 

[StrangeCharm]

I'm almost done with the problem but am having trouble with integrating 4/[x^(2)+4]. It looks like integrating 1/(1+x^2)dx and getting arctan(x); however, I'm not sure how to simplify the fraction into a similar form.
Also, I'm wondering whether it is valid to make ∫4/(4+x^2)dx ⇒ ∫[2/(2+x)]^2dx and using u-substitution with u=x+2. I'm not sure because doing this would get a different result from the method above.

 

[SammyS]

I'm almost done with the problem but am having trouble with integrating 4/[x^(2)+4]. It looks like integrating 1/(1+x^2)dx and getting arctan(x); however, I'm not sure how to simplify the fraction into a similar form.
Also, I'm wondering whether it is valid to make ∫4/(4+x^2)dx ⇒ ∫[2/(2+x)]^2dx and using u-substitution with u=x+2. I'm not sure because doing this would get a different result from the method above.

Is (2 + x)² = 4 + x² ? No.

Use u substitution: u = 2x .

 

[StrangeCharm]

Okay, right... I just realized that. Thanks for the help!