Let f(x) = 1/(x^2 -7)
Then.
F(x) = intg f(x)
= intg 1/(x^2 -7)
But, we know that:
x^2 - 7 = (x-sqrt7)(x+sqrt7)
==> F(x) = intg 1/(x-sqrt7)(x+sqrt7)
Let us rewrite:
A/(x-sqrt7) + B/(x+sqrt7) = 1/(x-sqrt7)(x+sqrt7)
==> A(x+sqrt7) + B(x-sqrt7) = 1
==> Ax + Asqrt7 + Bx - Bsqrt7 = 1
==> (A+B)x + (A-B)sqrt7 = 1
==> A+B = 0==> A = -B
==> (A-B)sqrt7 = 1
==> (-2B)sqrt7 = 1
==> B = -1/2sqrt7
==> A = 1/2sqrt7
==> F(x) = intg 1/2sqrt7(x-sqrt7) - 1/2sqrt7(x+sqrt7)
= (1/2sqrt7)*log (x-sqrt7) - (1/2sqrt7)*log (x+sqrt7)
= (1/2sqrt7)*[log (x-sqrt7) - log (x+sqrt7)]
= (1/2sqrt7)*log [(x-sqrt7)/(x+sqrt7)]
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