How do you integrate #f(x)=(x-2)/((x^2+5)(x-3)(x-1))# using partial fractions?

Question

Paisley Johnson · Accepted Answer

The answer is #=1/12ln(∣x-1∣)+1/28ln(∣x-3∣)-5/84ln(x^2+5)- 4/(21sqrt5)arctan(x/sqrt5)+C#

Let's perform the decomposition into partial fractions

#(x-2)/((x-1)(x-3)(x^2+5))=A/(x-1)+B/(x-3)+(Cx+D)/(x^2+5)#

#=(A(x-3)(x^2+5)+B(x-1)(x^2+5)+(Cx+D)(x-1)(x-3))/((x-1)(x-3)(x^2+5))#

Therefore,

#x-2=A(x-3)(x^2+5)+B(x-1)(x^2+5)+(Cx+D)(x-1)(x-3)#

Let #x=1#, #=>#, #-1=-12A#, #=>#, #A=1/12#

Let #x=3#, #=>#, #1=28B#, #=>#, #B=1/28#

Coefficients of #x^3#, #=>#, #0=A+B+C#

#C=-5/42#

Let #x=0#, #=>#, #-2=-15A-5B+3D#

#3D=15A+5B-2=15/12+5/28-2=-4/7#

#D=-4/21#

Therefore,

#(x-2)/((x-1)(x-3)(x^2+5))=(1/12)/(x-1)+(1/28)/(x-3)+(-5/42x-4/21)/(x^2+5)#

#=(1/12)/(x-1)+(1/28)/(x-3)-1/42(5x+8)/(x^2+5)#

So,

#I=int((x-2)dx)/((x-1)(x-3)(x^2+5))=1/12int(dx)/(x-1)+1/28int(dx)/(x-3)-1/42int((5x+8)dx)/(x^2+5)#

#=1/12ln(∣x-1∣)+1/28ln(∣x-3∣)-1/42(5int(xdx)/(x^2+5)+8int(dx)/(x^2+5))#

Let #u=x^2+5#, #=>#, #du=2xdx#

#int(xdx)/(x^2+5)=1/2 int (du)/u=1/2lnu=1/2ln(∣x^2+5∣)#

Let #v=x/sqrt5#, #=>#, #dv=dx/sqrt5#

#8int(dx)/(x^2+5)=8/sqrt5int(dv)/(v^2+1)#

Let #v=tantheta#, #dv=sec^2theta d theta#

and #1+tan^2theta=sec^2theta#

So,

#8/sqrt5int(dv)/(v^2+1)=8/sqrt5int(sec^2theta/sec^2thetad theta)#

#=8/sqrt5theta=8/sqrt5arctanv=8/sqrt5arctan(x/sqrt5)#

Putting it all together

#I=1/12ln(∣x-1∣)+1/28ln(∣x-3∣)-5/84ln(x^2+5)- 4/(21sqrt5)arctan(x/sqrt5)+C#

Evelyn Agard · Answer

undefined To integrate the function $ f(x) = \frac{x - 2}{(x^2 + 5)(x - 3)(x - 1)} $ using partial fractions, follow these steps:

1. Factor the denominator $ (x^2 + 5)(x - 3)(x - 1) $.
2. Decompose the fraction into partial fractions.
3. Solve for the constants.
4. Integrate each partial fraction separately.
5. Combine the integrals to find the overall integral of $ f(x) $.

Step 1: Factor the denominator:
$$ (x^2 + 5)(x - 3)(x - 1) = (x^2 + 5)(x^2 - 4x + 3) $$

Step 2: Decompose into partial fractions:
$$ \frac{x - 2}{(x^2 + 5)(x - 3)(x - 1)} = \frac{A}{x^2 + 5} + \frac{B}{x - 3} + \frac{C}{x - 1} $$

Step 3: Solve for the constants A, B, and C.
$$ x - 2 = A(x - 3)(x - 1) + B(x^2 + 5)(x - 1) + C(x^2 + 5)(x - 3) $$

Step 4: Solve for A, B, and C by substituting appropriate values of x.
$$ A = \frac{3}{32}, \quad B = -\frac{1}{8}, \quad C = \frac{1}{32} $$

Step 5: Integrate each partial fraction separately:
$$ \int \frac{A}{x^2 + 5} \, dx = \frac{A}{\sqrt{5}} \arctan{\left(\frac{x}{\sqrt{5}}\right)} + C_1 $$
$$ \int \frac{B}{x - 3} \, dx = B \ln{|x - 3|} + C_2 $$
$$ \int \frac{C}{x - 1} \, dx = C \ln{|x - 1|} + C_3 $$

Step 6: Combine the integrals and constants:
$$ \int \frac{x - 2}{(x^2 + 5)(x - 3)(x - 1)} \, dx = \frac{3}{32\sqrt{5}} \arctan{\left(\frac{x}{\sqrt{5}}\right)} - \frac{1}{8} \ln{|x - 3|} + \frac{1}{32} \ln{|x - 1|} + C $$

Where $ C $ is the constant of integration.