Partial Fractions

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Evaluate the integral $\int \frac{4x^2 + 3x + 5}{(x-1)(x^2+4)} \, dx$ .

\frac{12}{5}\ln|x-1| + \frac{4}{5}\ln(x^2+4) + C

\frac{12}{5}\ln|x-1| + \frac{8}{5}\ln(x^2+4) + \frac{23}{5}\arctan\left(\frac{x}{2}\right) + C

\frac{12}{5}\ln|x-1| + \frac{4}{5}\ln(x^2+4) - \frac{23}{10}\arctan\left(\frac{x}{2}\right) + C

\frac{12}{5}\ln|x-1| + \frac{4}{5}\ln(x^2+4) + \frac{23}{10}\arctan\left(\frac{x}{2}\right) + C

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Use partial fractions to evaluate the indefinite integral: $\int \frac{2x+5}{x^3+2x^2+x} \, dx$ .

5\ln|x| -5\ln|x+1| + \frac{3}{x+1} + C

5\ln|x| -5\ln|x+1| + \frac{3}{(x+1)^2} + C

5\ln|x| -2\ln|x+1| + \frac{3}{x+1} + C

5\ln|x| -5\ln|x+1| - \frac{3}{x+1} + C

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The rate of a chemical reaction (in moles per minute) is given by $R(t) = \frac{2t+3}{t^2+2t+1}$ . Find the total number of moles produced from $t=1$ to $t=3$ minutes.

2\ln 4 - \frac{1}{4}

2\ln 2 + \frac{1}{4}

moles

2\ln 2 - \frac{1}{4}

2\ln 2 + \frac{1}{2}

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Partial Fractions

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