Jawaban:
dgn cara integral rasional:
[tex] \frac{x - 7}{ {x}^{2} - x - 12 } [/tex]
[tex] = \frac{x - 7}{(x - 4)(x + 3)} [/tex]
[tex]mis. \: \frac{x - 7}{(x - 4)(x + 3)} = \frac{a}{x - 4} + \frac{b}{x + 3} [/tex]
[tex] \frac{x - 7}{(x - 4)(x + 3)} = \frac{a(x + 3) + b(x - 4)}{(x - 4)(x + 3)}[/tex]
[tex] \frac{x - 7}{(x - 4)(x + 3)} = \frac{(a + b)x+ (3a - 4b)}{(x - 4)(x + 3)}[/tex]
setarakan ruas kiri & ruas kanan, maka:
a+b = 1 <=> 3a + 3b = 3
3a-4b = -7 <=> 3a - 4b = -7
-------------------- (-)
7b = 10
b = 10/7
a = -3/7
[tex]int( \frac{x - 7}{(x - 4)(x + 3)} )=int( \frac{ - \frac{3}{7} }{x - 4}) + int( \frac{ \frac{10}{7} }{x + 3} )[/tex]
[tex] = - \frac{ 3}{7} int( \frac{1}{x - 4} )+ \frac{10}{7} int( \frac{1}{x + 3} )[/tex]
[tex] = - \frac{ 3}{7} ln(x - 4 )+ \frac{10}{7} ln(x + 3) + c[/tex]
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