Find int frac{x^{2}+1}{x^{2}-5 x+6} d x | vedclass

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Question

(A) The integrand $\frac{x^{2}+1}{x^{2}-5 x+6}$ is an improper rational function. By performing long division,we get:$\frac{x^{2}+1}{x^{2}-5 x+6} = 1

Vedclass · Accepted Answer

$x-5 \log |x-2|+10 \log |x-3|+C$

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(A) The integrand $\frac{x^{2}+1}{x^{2}-5 x+6}$ is an improper rational function. By performing long division,we get:$\frac{x^{2}+1}{x^{2}-5 x+6} = 1 + \frac{5x-5}{x^{2}-5x+6} = 1 + \frac{5x-5}{(x-2)(x-3)}$Using partial fractions for $\frac{5x-5}{(x-2)(x-3)}$:$\frac{5x-5}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}$$5x-5 = A(x-3) + B(x-2)$Setting $x=2$: $5(2)-5 = A(2-3) \implies 5 = -A \implies A = -5$Setting $x=3$: $5(3)-5 = B(3-2) \implies 10 = B$Thus,$\frac{x^{2}+1}{x^{2}-5 x+6} = 1 - \frac{5}{x-2} + \frac{10}{x-3}$Integrating both sides:$\int \left( 1 - \frac{5}{x-2} + \frac{10}{x-3} \right) dx = \int dx - 5 \int \frac{1}{x-2} dx + 10 \int \frac{1}{x-3} dx$$= x - 5 \log |x-2| + 10 \log |x-3| + C$

Find $\int \frac{x^{2}+1}{x^{2}-5 x+6} d x$

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