Indefinite Integral with Natural Log?

Question

If I take a problem like ∫ x∕x²-3 dx and that the integral is ½ln|x²-3|+C, what I can't figure out is why I have to multiply by 2 on the right side and divide by 2 on the left?  Why am I not doing the same thing to both sides of the integral like I would an equation?

Null · Answer

∫ x∕x²-3 dx , well

the derivative of ln(f(x)) is  f'(x) / f(x) , where f(x) differs from 0, hence we would have taken an integral easily if the numerator was the derivative of denominator,

observe, numerator is x , and denominator is x^2-3, and the derivative of denominator is 2x, so numerator is not equal to derivative of denominator, but if numerator was 2x , then it would exactly be the derivative of the denominator, so

make it be, multiply and divide by 2, then you would get,

∫ x ∕ (x²-3) dx = (1/2) ∫ (2x) ∕ (x²-3) dx , so

= (1/2)ln|x^2-3| + c

Indefinite Integral with Natural Log?

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