How do you integrate #(2x^2+4x+12)/(x^2+7x+10)# using partial fractions?

Answer 1

The answer is #=2x-14ln(|x+5|)+4ln(|x+2|)+C#

The numerator is

#2x^2+4x+12=2(x^2+2x+6)#

We perform a long division

#color(white)(aaaa)##x^2+2x+6##color(white)(aaaaa)##|##x^2+7x+10#
#color(white)(aaaa)##x^2+7x+10##color(white)(aaaa)##|##1#
#color(white)(aaaaaa)##0-5x-4#

Therefore

#(2x^2+4x+12)/(x^2+7x+10)=2(1-(5x+4)/(x^2+7x+10))#

We factorise the denominator

#x^2+7x+10=(x+5)(x+2)#

We can perform the decomposition into partial fractions

#(5x+4)/(x^2+7x+10)=(5x+4)/((x+5)(x+2))#
#=A/(x+5)+B/(x+2)=(A(x+2)+B(x+5))/((x+5)(x+2))#

The denominators are the same, we compare the numerators

#5x+4=A(x+2)+B(x+5)#
Let #x=-5#, #=>#, #-21=-3A#, #=>#, #A=7#
Let #x=-2#, #=>#, #-6=3B#, #=>#, #B=-2#

So,

#(2x^2+4x+12)/(x^2+7x+10)=2(1-(7/(x+5)-2/(x+2)))#

Therefore,

#int((2x^2+4x+12)dx)/(x^2+7x+10)=2int(1-(7/(x+5)-2/(x+2)))dx#
#=2intdx-14intdx/(x+5)+4intdx/(x+2)#
#=2x-14ln(|x+5|)+4ln(|x+2|)+C#
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Answer 2

To integrate the rational function ( \frac{2x^2 + 4x + 12}{x^2 + 7x + 10} ) using partial fractions, follow these steps:

  1. Factor the denominator (x^2 + 7x + 10): (x^2 + 7x + 10 = (x + 5)(x + 2)).
  2. Decompose the rational function into partial fractions: [ \frac{2x^2 + 4x + 12}{x^2 + 7x + 10} = \frac{A}{x + 5} + \frac{B}{x + 2} ]
  3. Multiply both sides by the denominator (x^2 + 7x + 10) to clear the fractions: [ 2x^2 + 4x + 12 = A(x + 2) + B(x + 5) ]
  4. Expand and collect like terms: [ 2x^2 + 4x + 12 = (A + B)x + 2A + 5B ]
  5. Equate coefficients of corresponding terms: [ A + B = 2 ] [ 2A + 5B = 12 ]
  6. Solve the system of equations to find the values of (A) and (B).
  7. Once you have (A) and (B), integrate each term separately: [ \int \frac{2x^2 + 4x + 12}{x^2 + 7x + 10} dx = \int \frac{A}{x + 5} dx + \int \frac{B}{x + 2} dx ] [ = A\ln|x + 5| + B\ln|x + 2| + C ]
  8. Substitute the values of (A) and (B) into the integral.
  9. Finally, add the constant of integration (C) if necessary.
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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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