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Step-by-step explanation:

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Answer:

[tex]\int {\frac{2x-3}{x^{2} +4x-12} } \, dx[/tex] =  [tex]\frac{1}{4}[/tex] * log (x+6)⁹ (x-2)

Step-by-step explanation:

Given that [tex]\int {\frac{2x-3}{x^{2} +4x-12} } \, dx[/tex]

We factorize the denominator that is  x² + 4x - 12

We know that 6 + (-2) = 4 and 6*(-2) = -12

Therefore, by mid term splitting, we get,

x² + 4x - 12 = x² + 6x - 2x - 12

                  = x(x+6) -2(x+6)

                  = (x+6) (x - 2)

=>   [tex]\int {\frac{2x-3}{x^{2} +4x-12} } \, dx[/tex] =  [tex]\int {\frac{2x-3}{(x+6)(x-2)} } \, dx[/tex]

Now as the denominator as two not repeated factors,

We integrate using partial fractions,

=> [tex]\frac{2x-3}{(x+6)(x-2)}[/tex] = [tex]\frac{A}{x+6}[/tex] + [tex]\frac{B}{x-2}[/tex]

                    = [tex]\frac{A(x-2) + B(x+6)}{(x+6)(x-2)}[/tex]

=> 2x - 3 = Ax - 2A + Bx + 6B

Equating the terms on both the sides, we get,

2x = Ax - Bx  and -3 = -2A + 6B

=> 2 = A - B      and  - 3 = -2A + 6B

=> A = 2 + B

We will substitute this value in the other equation,

=>  - 3 = -2(2+B) + 6B

=> -3 = -4 -2B + 6B

=> -3+4 = 4B

=> 1 = 4B

=> B = 1/4

As A = 2+B

=> A = 2 + 1/4 = 9/4

We substitute these values in the equation.

=> [tex]\frac{2x-3}{(x+6)(x-2)}[/tex] = [tex]\frac{A}{x+6}[/tex] + [tex]\frac{B}{x-2}[/tex] = [tex]\frac{9}{4(x+6)}[/tex] + [tex]\frac{1}{4(x-2)}[/tex]

Integrating now,

[tex]\int {\frac{2x-3}{(x+6)(x-2)} } \, dx[/tex] = [tex]\int {\frac{9}{4(x+6)} } \, dx[/tex] + [tex]\int {\frac{1}{4(x-2)} } \, dx[/tex]

                       = [tex]\frac{9}{4}\int {\frac{1}{(x+6)} } \, dx[/tex] +   [tex]\frac{1}{4}\int {\frac{1}{(x-2)} } \, dx[/tex]

                      = [tex]\frac{9}{4}[/tex] log |x+6| +    [tex]\frac{1}{4}[/tex] log |x-2|

                      = [tex]\frac{1}{4}[/tex] * (9log |x+6| + log |x-2|)

                      = [tex]\frac{1}{4}[/tex] * (log |x+6|⁹ + log |x-2|)      (as log mⁿ = n log m)

                      = [tex]\frac{1}{4}[/tex] * log (x+6)⁹ (x-2)                (as log m + log n = log mn)

Therefore, [tex]\int {\frac{2x-3}{x^{2} +4x-12} } \, dx[/tex] =  [tex]\frac{1}{4}[/tex] * log (x+6)⁹ (x-2)