# How do you integrate #f(x)=(3x-4)/(2x+1)# using the quotient rule?

##### 1 Answer

Dec 20, 2016

# int f(x) dx= 3/2x-11/4ln|2x+1| + C#

#### Explanation:

There is no such thing as a quotient rule for integration!

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

To integrate this particular function we can use the following substitution:

#{: ( "Let ", u=2x+1, => x=1/2u-1/2, => 3x-4=3/2u-11/2 ), ("Then ", (du)/dx=2, => dx=1/2du,) :}#

And so:

# int (3x-4)/(2x+1) dx = int (3/2u-11/2)/u 1/2du#

# " " = 1/4int (3u-11)/u du#

# " " = 1/4int 3-11/u du#

# " " = 1/4{3u - 11ln|u|} + C'#

# " " = 1/4{3(2x+1) - 11ln|2x+1|} + C'#

# " " = 1/4{6x+3 - 11ln|2x+1|} + C'#

# " " = 3/2x+3/4-11/4ln|2x+1|} + C'#

# " " = 3/2x-11/4ln|2x+1| + C#