Evaluate the Integral. Show All Steps

Question

Evaluate the Integral. Show All Steps. If possible attach a video going through the problem

Answer 1

First step is to do long division and get

2 + [ (-x^2 +4x+7)/(x^3-x^2+4x-4)]

Next factor the denominator of the term on the right by noticing that x=1 is a root of x^3-x^2+4x-4, so it is divisible by (x-1). Perform long division to rewrite the denominator as (x-1)(x^2+4). Thus the integrate is now

2 + [ (-x^2 +4x+7)/((x-1)(x^2 +4))]. Next, use partial fractions to rewrite

[(-x^2 +4x+7)/((x-1)(x^2 +4)) = A/(x-1) + (Bx+C)/(x^2 + 4)

and take a common denominator on the right to get A=2, B=-3 and C=1. So now the integrand is

2 +. 2/(x-1) + (-3x+1)/(x^2 +4)

= 2 + 2/(x-1) - 3x/(x^2+4) + 1/(x^2 +4)

integrate this with respect to x to get

2x + 2 ln|x-1| - (3/2)ln (x^2 + 4) + (1/2) arctan (x/2) + C