How do you integrate #(x+7)/(x^2(x+2))# using partial fractions?

Question

Isaiah Allen · Accepted Answer

The answer is #=-7/(2x)-5/4ln(|x|)+5/4ln(|x+2|)+C#

Let's perform the decomposition into partial fractions

#(x+7)/(x^2(x+2))=A/(x^2)+B/(x)+C/(x+2)#

#=(A(x+2)+B(x(x+2))+C(x^2))/(x^2(x+2))#

As the denominators are the same, we compare the numerators

#x+7=A(x+2)+B(x(x+2))+C(x^2)#

Let #A=0#, #=>#, #7=2A#, #=>#, #A=7/2#

Let #x=-2#, #=>#, #5=4C#, #=>#, #C=5/4#

Coefficients of #x#

#1=A+2B#, #=>#, #2B=1-A=1-7/2=-5/2#

#B=-5/4#

Therefore,

#(x+7)/(x^2(x+2))=(7/2)/(x^2)+(-5/4)/(x)+(5/4)/(x+2)#

So,

#int((x+7)dx)/(x^2(x+2))=7/2intdx/(x^2)-5/4intdx/(x)+5/4intdx/(x+2)#

#=-7/(2x)-5/4ln(|x|)+5/4ln(|x+2|)+C#

Emilia Aguilar · Answer

undefined To integrate (x+7)/(x^2(x+2)) using partial fractions, first express the fraction in the form of partial fractions. Then integrate each term separately. Here's how you proceed:

1. Decompose the fraction (x+7)/(x^2(x+2)) into partial fractions:
   (x+7)/(x^2(x+2)) = A/x + B/x^2 + C/(x+2)

2. Multiply both sides by the denominator x^2(x+2) to clear the fractions:
   x + 7 = A(x+2) + B(x)(x+2) + C(x^2)

3. Equate coefficients to find the values of A, B, and C.

4. Once you find the values of A, B, and C, rewrite the original integral as:
   ∫(A/x + B/x^2 + C/(x+2)) dx

5. Integrate each term separately:
   ∫A/x dx + ∫B/x^2 dx + ∫C/(x+2) dx

6. Integrate each term using the power rule for integration.

7. Finally, combine the results to get the integral of the original function.

Please note that the specific values of A, B, and C depend on the coefficients of the original expression.

Isabella Ackerman · Answer

undefined To integrate $ \frac{x+7}{x^2(x+2)} $ using partial fractions, you first decompose the fraction into simpler fractions. The partial fraction decomposition will be of the form $ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+2} $. After finding the values of $ A $, $ B $, and $ C $, you integrate each term individually.

To find the values of $ A $, $ B $, and $ C $, you typically clear the denominators by multiplying both sides of the equation by the common denominator $ x^2(x+2) $, and then equate coefficients of like terms.

Once you have the values of $ A $, $ B $, and $ C $, you integrate each term separately. The integral of $ \frac{A}{x} $ is $ A\ln|x| $, the integral of $ \frac{B}{x^2} $ is $ -\frac{B}{x} $, and the integral of $ \frac{C}{x+2} $ is $ C\ln|x+2| $.

After integrating each term, you combine them to get the final result.