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int(x^(2)-5x+6)/(2x-6)dx=

∫x2−5x+62x−6dx= \int \frac{x^{2}-5 x+6}{2 x-6} d x=

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Evaluate definite integrals using the power ruleright chevron icon

Full solution

Q. ∫x2−5x+62x−6dx= \int \frac{x^{2}-5 x+6}{2 x-6} d x=
  1. Factorize Numerator: Simplify the integral by factoring the numerator.\newlineNumerator: x2−5x+6=(x−2)(x−3)x^2 - 5x + 6 = (x-2)(x-3).\newlineDenominator: 2x−6=2(x−3)2x - 6 = 2(x-3).\newlineIntegral becomes: ∫(x−2)(x−3)2(x−3)dx\int \frac{(x-2)(x-3)}{2(x-3)} dx.
  2. Cancel Common Terms: Cancel out common terms in the numerator and the denominator.\newline(x−2)(x−3)2(x−3)=x−22\frac{(x-2)(x-3)}{2(x-3)} = \frac{x-2}{2} when x≠3x \neq 3.\newlineIntegral simplifies to: ∫x−22dx\int \frac{x-2}{2} dx.
  3. Integrate Simplified Expression: Integrate the simplified expression.\newline∫x−22dx=12∫(x−2)dx\int \frac{x-2}{2} dx = \frac{1}{2} \int (x-2) dx.\newline12∫(x−2)dx=12(x22−2x)+C\frac{1}{2} \int (x-2) dx = \frac{1}{2} \left( \frac{x^2}{2} - 2x \right) + C.

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