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Integration of Rational Functions


Given \(\displaystyle{f(x) = \int \frac{2x + 49}{x^2+9x+14} dx}\) and \(f(0) = 1,\) find the function \(f(x).\)

Evaluate \[\int_{1}^{8} \frac{4}{x^3} dx .\]

If \(\displaystyle{f(x) = \int \frac{8x-9}{(x-3)(x-8)} dx},\) what is \(f(5)-f(1)?\)

If \(\displaystyle{f(x)=\int \frac{38x-2}{19x^2-2x-2e}dx}\) and \(f(0)=1+\ln 2-\ln 19,\) what is the value of \(f(-e)?\)

Evaluate the indefinite integral \[\int \frac{x^2+17}{x-3}dx.\] (Use \(C\) as the constant of integration.)


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