Question

Integral of (x+3)/(2x^3-8x)

Answer 1

Step-by-step explanation:

int: (x + 3) / (2x * (x + 2) * (x - 2))

By partial fractions, we set up the following:

A/2x + B/(x + 2) + C/(x - 2)

Combine and set equal to the numerator:

A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2) = x + 3

Let x = -2, solve for B:

0 + 16B + 0 = 1

B = 1/16

Let x = 0, solve for A:

-4A + 0 + 0 = 3

A = -3/4

Let x = 2, solve for C:

0 + 0 + 16C = 5

C = 5/16

Now, we have our new integral:

int: ((-3/4) / 2x) + ((1/16) / (x + 2)) + ((5/16) / (x - 2))

All done with natural logs, to get:

(-3/8)ln(2x) + (1/16)ln(x + 2) + (5/16)ln(x - 2) + C