How do you integrate #int (x^3 + 2) / (x^4 -16)# using partial fractions?

Question

Luke Alvarez · Accepted Answer

#3/16 ln abs(x+2) + 5/16 ln abs(x-2) + 1/4 ln(x^2+4) - 1/8 arctan (x/2) + K#

Firstly, you need the #dx# at the end of the integral for formal presentation i.e. #int (x^3 + 2)/(x^4-16) dx#.

Secondly, you need to decompose #(x^3 + 2)/(x^4-16)# into partial fractions. This is made simpler by noting that

#x^4-16 = (x^2-4)(x^2+4) = (x+2)(x-2)(x^2+4)#

That is,

#(x^3 + 2)/(x^4-16) = A/(x+2) + B/(x-2) + (Cx+D)/(x^2+4)#

for real constants #A#, #B#, #C# and #D#.

Using the cover-up rule and comparison of coefficients by algebraic means, you should be able to find solve for the unknowns, that is, #A = 3/16#, #B = 5/16#, #C=1/2# and #D=–1/4#.

Thus, #(x^3 + 2)/(x^4-16) = (3/16)/(x+2) + (5/16)/(x-2) + (1/2x-1/4)/(x^2+4)#, and

#int (x^3 + 2)/(x^4-16) dx=int ((3/16)/(x+2) + (5/16)/(x-2) + (1/2x-1/4)/(x^2+4)) dx#.

The first two integrals are easy to understand.

#int (3/16)/(x+2) dx = 3/16 int 1/(x+2) dx = 3/16 ln abs(x+2)+K_1#

and similarly,

#int (5/16)/(x+2) dx = 5/16 ln abs(x-2)+K_2#

The third section is a little more complex. Through suitable decompositions,

#int (1/2x-1/4)/(x^2+4) dx = 1/4 int (2x)/(x^2+4) dx - 1/4 int 1/(x^2+4) dx = 1/4 ln(x^2+4) - 1/4(1/2)arctan (x/2) + K_3 = 1/4 ln(x^2+4) - 1/8 arctan (x/2) + K_3#

Consequently, combining everything,

#int (x^3 + 2)/(x^4-16) dx=3/16 ln abs(x+2) + 5/16 ln abs(x-2) + 1/4 ln(x^2+4) - 1/8 arctan (x/2) + K#

where #K = K_1 + K_2 + K_3#.

Isaiah Allen · Answer

undefined To integrate $ \int \frac{x^3 + 2}{x^4 - 16} $ using partial fractions, follow these steps:

1. Factor the denominator $ x^4 - 16 $ into irreducible factors.
2. Write the fraction $ \frac{x^3 + 2}{x^4 - 16} $ as a sum of partial fractions.
3. Determine the unknown coefficients.
4. Integrate each term of the partial fractions separately.

Let's proceed:

1. $ x^4 - 16 $ factors as $ (x^2 + 4)(x^2 - 4) $.
2. Write the fraction as $ \frac{x^3 + 2}{(x^2 + 4)(x + 2)(x - 2)} $.
3. Express the fraction as the sum of partial fractions: $ \frac{Ax + B}{x^2 + 4} + \frac{C}{x + 2} + \frac{D}{x - 2} $.
4. Determine the unknown coefficients $ A $, $ B $, $ C $, and $ D $.
5. After finding the values of $ A $, $ B $, $ C $, and $ D $, integrate each term separately.
6. Perform the integration of each term, then add the results together to obtain the final answer.