Help Partial fraction

[Yelan]

Integrate the partial fraction. 4x² + 8x -12 ÷ x³ +7x² +12x

 

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To integrate the given expression using partial fraction decomposition, we first need to factorise the denominator:

x³ +7x² +12x = x(x²+7x+12) = x(x+3)(x+4)

So, we can write the given expression as:

(4x² + 8x -12) ÷ x(x+3)(x+4)

Now, we need to find the partial fraction decomposition:

(4x² + 8x -12) ÷ x(x+3)(x+4) = A ÷ x + B ÷ (x+3) + C ÷ (x+4)

Multiplying through by the common denominator, we get:

4x² + 8x -12 = A(x+3)(x+4) + Bx(x+4) + Cx(x+3)

Simplifying and equating coefficients, we get:

4x² + 8x -12 = (A+B+C)x² + (7A+4B+3C)x + 12A

Comparing coefficients of x², x, and the constant term, we get:

A+B+C = 0 ...(1)
7A+4B+3C = 8 ...(2)
12A = -12 ...(3)

From equation (3), we get A = -1. Substituting this value in equation (1), we get B+C = 1.

From equation (2), we get 7A+4B+3C = 8. Substituting A = -1, we get -7 + 4B + 3C = 8, or 4B + 3C = 15.

Solving these equations simultaneously, we get B = 4 and C = -3.

Therefore, the given expression can be written as:

4x² + 8x -12 = (-1) ÷ x + 4 ÷ (x+3) - 3 ÷ (x+4)

Integrating each term individually, we get:

∫(4x² + 8x -12) ÷ x(x+3)(x+4) dx = ∫(-1) ÷ x dx + ∫4 ÷ (x+3) dx - ∫3 ÷ (x+4) dx

= -ln|x| + 4ln|x+3| - 3ln|x+4| + C

where C is the constant of integration.

 

[Yelan]

To integrate the given expression using partial fraction decomposition, we first need to factorise the denominator:

x³ +7x² +12x = x(x²+7x+12) = x(x+3)(x+4)

So, we can write the given expression as:

(4x² + 8x -12) ÷ x(x+3)(x+4)

Now, we need to find the partial fraction decomposition:

(4x² + 8x -12) ÷ x(x+3)(x+4) = A ÷ x + B ÷ (x+3) + C ÷ (x+4)

Multiplying through by the common denominator, we get:

4x² + 8x -12 = A(x+3)(x+4) + Bx(x+4) + Cx(x+3)

Simplifying and equating coefficients, we get:

4x² + 8x -12 = (A+B+C)x² + (7A+4B+3C)x + 12A

Comparing coefficients of x², x, and the constant term, we get:

A+B+C = 0 ...(1)
7A+4B+3C = 8 ...(2)
12A = -12 ...(3)

From equation (3), we get A = -1. Substituting this value in equation (1), we get B+C = 1.

From equation (2), we get 7A+4B+3C = 8. Substituting A = -1, we get -7 + 4B + 3C = 8, or 4B + 3C = 15.

Solving these equations simultaneously, we get B = 4 and C = -3.

Therefore, the given expression can be written as:

4x² + 8x -12 = (-1) ÷ x + 4 ÷ (x+3) - 3 ÷ (x+4)

Integrating each term individually, we get:

∫(4x² + 8x -12) ÷ x(x+3)(x+4) dx = ∫(-1) ÷ x dx + ∫4 ÷ (x+3) dx - ∫3 ÷ (x+4) dx

= -ln|x| + 4ln|x+3| - 3ln|x+4| + C

where C is the constant of integration.

Thank you

 

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