How do you integrate #(3x^2+9x-4)/((x-1)(x^2+4x-1))# using partial fractions?

Answer 1

Please see the explanation.

Find the partial fractions.

#(3x^2+9x-4)/((x-1)(x^2+4x-1)) = A/(x-1)+(Bx+C)/(x^2+4x-1)#
#(3x^2+9x-4) = A(x^2+4x-1)+(Bx+C)(x-1)#

Eliminate B and C by letting x = 1:

#(3(1)^2+9(1)-4) = A((1)^2+4(1)-1)#
#8=A4#
#A = 2#
#(3x^2+9x-4) = 2(x^2+4x-1)+(Bx+C)(x-1)#

Eliminate B by letting x = 0:

#(3(0)^2+9(0)-4) = 2((0)^2+4(0)-1)+C((0)-1)#
#-4=-2-C#
#C = 2#
#(3x^2+9x-4) = 2(x^2+4x-1)+(Bx+2)(x-1)#

Let x = -1:

#(3(-1)^2+9(-1)-4) = 2((-1)^2+4(-1)-1)+(B(-1)+2)(-1-1)#
#-10=-8+2B-4#
#B = 1#
#int(3x^2+9x-4)/((x-1)(x^2+4x-1))dx = 2int1/(x-1)dx+int(x+2)/(x^2+4x-1)dx#

Multiply the second integral by 1/2 so that the numerator can be multiplied by 2:

#int(3x^2+9x-4)/((x-1)(x^2+4x-1))dx = 2int1/(x-1)dx+1/2int(2x+4)/(x^2+4x-1)dx#

Both integrals become the natural logarithm:

#int(3x^2+9x-4)/((x-1)(x^2+4x-1))dx = 2ln|x-1|+1/2ln|x^2+4x-1|+C#
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Answer 2

To integrate (\frac{{3x^2+9x-4}}{{(x-1)(x^2+4x-1)}}) using partial fractions, follow these steps:

  1. Factor the denominator ( (x-1)(x^2+4x-1) ) if necessary.
  2. Express the given fraction as the sum of two or more fractions with simpler denominators.
  3. Set up equations to find the unknown coefficients.
  4. Solve the equations to find the values of the coefficients.
  5. Integrate each partial fraction separately.

Step 1: Factor the denominator: (x^2+4x-1) does not factor easily using integers, so it remains as is.

Step 2: Express the given fraction as partial fractions: (\frac{{3x^2+9x-4}}{{(x-1)(x^2+4x-1)}} = \frac{A}{x-1} + \frac{Bx + C}{x^2+4x-1})

Step 3: Set up equations: [3x^2 + 9x - 4 = A(x^2 + 4x - 1) + (Bx + C)(x - 1)]

Step 4: Solve for (A), (B), and (C): Expand and match coefficients to solve for (A), (B), and (C).

Step 5: Integrate each partial fraction separately: [ \int \frac{A}{x-1} , dx + \int \frac{Bx + C}{x^2+4x-1} , dx ]

[ = A \ln|x-1| + \int \frac{Bx + C}{x^2+4x-1} , dx ]

[ = A \ln|x-1| + \int \frac{Bx + C}{(x+2)^2 - 5} , dx ]

After this point, you would integrate the second term using substitution or partial fractions again, depending on the complexity of the integral. Once integrated, you would have your final solution.

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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.

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