How do you integrate #int(x)/((3x-2)(x+2)(2x-1))# using partial fractions?

Answer 1

#1/4ln|3x-2| - 1/20ln|x+2| -1/5ln|2x-1| + c#

Since the factors on the denominator are linear , the numerators will be constants , say A , B and C.

#x/((3x-2)(x+2)(2x-1)) = A/(3x-2) + B/(x+2) + C/(2x-1)#

Now multiply through by (3x-2)(x+2)(2x-1)

hence : x = A(x+2)(2x-1) + B(3x-2)(2x-1) + C(3x-2)(x+2)...(1)

The aim now is to find values of A , B and C. Note that if x =-2 , the terms with A and C will be zero. If x#=1/2" the terms with A and B will be zero"# and if x#=2/3"the terms with B and C will be zero"#
let x = -2 in (1): - 2 = 40B #rArr B =-1/20# let #x = 1/2" in (1)" : 1/2 = -5/4C rArr C= -2/5 # let # x = 2/3 " in (1)": 2/3 = 8/9A rArr A = 3/4 #

hence integral becomes

#int(3/4)/(3x-2) dx -(1/20)/(x+2) dx -(2/5)/(2x-1) dx #
# = 3/4 . 1/3 ln|3x-2| -1/20ln|x+2| -2/5 . 1/2 ln|2x-1| + c#
# = 1/4ln|3x-2| - 1/20ln|x+2| - 1/5ln|2x-1| + c#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To integrate ( \frac{x}{(3x-2)(x+2)(2x-1)} ) using partial fractions, you first decompose the fraction into partial fractions. After decomposing, you equate the original fraction to the sum of the partial fractions and then solve for the constants. Once you have the partial fractions, you integrate each term separately.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7