How do you integrate #int(x+1)/((x+5)(x+2)(x-5))# using partial fractions?

Question

Charles Anctil · Accepted Answer

The answer is #=-2/15(∣x+5∣)+1/21ln(∣x+2∣)+3/35ln(∣x-5∣)+C#

Perform the decomposition into partial fractions

#(x+1)/((x+5)(x+2)(x-5))=A/(x+5)+B/(x+2)+C/(x-5)#

#=(A(x+2)(x-5)+B(x+5)(x-5)+C(x+5)(x+2))/((x+5)(x+2)(x-5))#

Therefore, by equating the numerators

#(x+1)=(A(x+2)(x-5)+B(x+5)(x-5)+C(x+5)(x+2))#

Let #x=-5#, #=>#, #-4=30A#, #=>#, #A=-2/15#

Let #x=-2#, #=>#, #-1=-21B#, #=>#, #B=1/21#

Let #x=5#, #=>#, #6=70C#, #=>#, #C=3/35#

so,

#(x+1)/((x+5)(x+2)(x-5))=(-2/15)/(x+5)+(1/21)/(x+2)+(3/35)/(x-5)#

Now, perform the integration

#int((x+1)dx)/((x+5)(x+2)(x-5))=-2/15intdx/(x+5)+1/21intdx/(x+2)+3/35intdx/(x-5)#

#=-2/15(∣x+5∣)+1/21ln(∣x+2∣)+3/35ln(∣x-5∣)+C#

Isabella Ackerman · Answer

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

1. Perform long division to express the fraction in the form $\frac{{x+1}}{{(x+5)(x+2)(x-5)}} = A(x) + \frac{{Bx + C}}{{x+5}} + \frac{{Dx + E}}{{x+2}} + \frac{{Fx + G}}{{x-5}}$.

2. Find the values of $A$, $B$, $C$, $D$, $E$, and $F$ by comparing coefficients.

3. Integrate each partial fraction term separately.

4. Combine the results to obtain the final integral.

Following these steps:

1. $x+1 = A(x+5)(x+2)(x-5) + (Bx + C)(x+2)(x-5) + (Dx + E)(x+5)(x-5) + (Fx + G)(x+5)(x+2)$.

2. After solving for $A$, $B$, $C$, $D$, $E$, and $F$, you'll find:
   $A = \frac{1}{49}$, $B = \frac{1}{49}$, $C = -\frac{7}{49}$, $D = \frac{5}{49}$, $E = \frac{13}{49}$, $F = \frac{7}{49}$, $G = -\frac{5}{49}$.

3. Integrating each term separately, you'll get:
   $\frac{1}{49} \ln|x+5| - \frac{1}{49} \ln|x+2| + \frac{5}{49} \ln|x-5| + \frac{7}{49} \ln|x+2| + \frac{7}{7} \ln|x-5| - \frac{5}{7} \ln|x+5|$.

4. Combine the results:
   $\frac{1}{49} \ln\left|\frac{(x+5)^2(x-5)}{(x+2)}\right| + \frac{7}{49} \ln\left|\frac{(x-5)^7}{(x+5)}\right| + C$.

Therefore, the integral of $\frac{{x+1}}{{(x+5)(x+2)(x-5)}}$ is $\frac{1}{49} \ln\left|\frac{(x+5)^2(x-5)}{(x+2)}\right| + \frac{7}{49} \ln\left|\frac{(x-5)^7}{(x+5)}\right| + C$, where $C$ is the constant of integration.