`int (4x^2+2x-1)/(x^3+x^2) dx` Use partial fractions to find the indefinite integral

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`int (4x^2+2x-1)/(x^3+x^2)dx`

To solve using partial fraction method, the denominator of the integrand should be factored.


Take note that if the factor in the denominator is linear and non-repeating, each factor in the denominator has a partial fraction form of `A/(ax+b)` .

And if the factor is linear and repeating, its partial fraction decomposition has a form `A_1/(ax+b) + A_2/(ax+b)^2+... +A_n/(ax+b)^n` .

So, expressing the integrand as sum of fractions, it becomes:


To determine the values of A, B and C, multiply both sides by the LCD of the fractions present.



Then, assign values to x in which either x, x^2 or x+1 will become zero.

So plug-in x=0 to get the value of B.


`-1=A(0)+ B(1)+C(0)`


Also, plug-in x=-1 to get the value of C.




To get the value of A, plug-in the values of B and C. Also, assign any value to x. Let it be x=1.

`4(1)^2+2(1)-1=A(1)(1+1)+ (-1)(1+1)+1(1)^2`





So, the partial fraction decomposition of the integrand is:


`= int (4x^2+2x-1)/(x^2(x+1))dx`

`= int (3/x -1/x^2+1/(x+1))dx`

Then, express it as three integrals.

`= int 3/x dx - int 1/x^2 dx + int 1/(x+1)dx`

`= 3int 1/x dx- int x^(-2) dx + int 1/(x+1)dx`

For the first and third integral, apply the formula `int 1/u du = ln|u|+C` .

And for the second integral, apply the formula `int u^n du = u^(n+1)/(n+1)+C` .

`= 3ln|x| + x^(-1) + ln|x+1| +C`

`=3ln|x| +1/x + ln|x+1| +C`


Therefore, `int(4x^2+2x-1)/(x^3+x^2)=3ln|x| +1/x + ln|x+1| +C` .

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial Team