How do you evaluate the integral #int (x^5+3x^2+1)/(x^4-1)dx#?

Question

Evelyn Agard · Accepted Answer

The answer is #=x^2/2-3/4ln(|x+1|)+5/4ln(|x-1|)-1/4ln(x^2+1)+arctanx+C#

We need

#intdx/(x^2+1)=arctan(x)#

As the degree of the numerator is greater than the degree of the denominator, perform a long division first.

#(x^5+3x^2+1)/(x^4-1)=x+(3x^2+x+1)/(x^4-1)#

Perform a decomposition into partial fractions

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

#=(Ax+B)/(x^2+1)+C/(x+1)+D/(x-1)#

#=((Ax+B)(x+1)(x-1)+C(x^2+1)(x-1)+D(x^2+1)(x+1))/((x^2+1)(x+1)(x-1))#

The denominators are the same , compare the numerators

#3x^2+x+1=(Ax+B)(x+1)(x-1)+C(x^2+1)(x-1)+D(x^2+1)(x+1)#

Let #x=1#, #=>#, #5=4D#, #=>#, #D=5/4#

Let #x=-1#, #=>#, #3=-4C#, #=>#, #C=-3/4#

Let #x=0#, #=>#, #1=-B-C+D#, #=>#, #B=5/4+3/4-1=1#

Coefficients of #x^3#

#0=A+C+D#, #=>#, #A=-C-D=3/4-5/4=-1/2#

Therefore,

#(x^5+3x^2+1)/(x^4-1)=x+(-1/2x+1)/(x^2+1)+(-3/4)/(x+1)+(5/4)/(x-1)#

So,

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

#intxdx=x^2/2#

#int(-3/4dx)/(x+1)=-3/4ln(|x+1|)#

#int(5/4dx)/(x-1)=5/4ln(|x-1|)#

#int((-1/2x+1)dx)/(x^2+1)=-1/4int(2xdx)/(x^2+1)+intdx/(x^2+1)=-1/4ln(x^2+1)+arctanx#

Finally,

#int((x^5+3x^2+1)dx)/(x^4-1)=x^2/2-3/4ln(|x+1|)+5/4ln(|x-1|)-1/4ln(x^2+1)+arctanx+C#

Samantha Adkison · Answer

undefined To evaluate the integral $\int \frac{x^5+3x^2+1}{x^4-1}dx$, we first perform polynomial long division to simplify the integrand. The result of the division is $x + \frac{3x^2}{x^4-1}$.

Next, we decompose $\frac{3x^2}{x^4-1}$ into partial fractions.

$$\frac{3x^2}{x^4-1} = \frac{3x^2}{(x^2+1)(x^2-1)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+1}$$

Solving for $A$, $B$, $C$, and $D$ yields $A = \frac{3}{2}$, $B = -\frac{3}{2}$, $C = 0$, and $D = 0$.

So, the integral becomes:

$$\int \left( x + \frac{3}{2(x-1)} - \frac{3}{2(x+1)} \right) dx$$

Integrating term by term, we get:

$$\frac{x^2}{2} + \frac{3}{2}\ln|x-1| - \frac{3}{2}\ln|x+1| + C$$

where $C$ is the constant of integration.