AP Calculus BC 6.12 Integrating Using Linear Partial Fractions Questions

Review partial fractions by decomposing rational functions into integrable terms before finding an antiderivative and checking the decomposition.

Covered syllabus points

FUN-6.F—For integrands requiring integration by linear partial fractions: (a) Determine indefinite integrals. (b) Evaluate definite…

[Maximum number: 3]

Consider the family of functions $f(x)=\frac{1}{x^{2}-2 x+k}$ , where k is a constant.

For k=-8, find the value of $\int_{0}^{1} f(x) d x$ .

\begin{aligned} & \frac{1}{x^{2}-2 x-8}=\frac{1}{(x-4)(x+2)}=\frac{A}{x-4}+\frac{B}{x+2} \\ & \Rightarrow 1=A(x+2)+B(x-4) \\ & \Rightarrow A=\frac{1}{6}, B=-\frac{1}{6} \end{aligned}

\begin{aligned} \int_{0}^{1} f(x) d x & =\int_{0}^{1}\left(\frac{\frac{1}{6}}{x-4}-\frac{\frac{1}{6}}{x+2}\right) d x \\ & =\left[\frac{1}{6} \ln |x-4|-\frac{1}{6} \ln |x+2|\right]_{x=0}^{x=1} \\ & =\left(\frac{1}{6} \ln 3-\frac{1}{6} \ln 3\right)-\left(\frac{1}{6} \ln 4-\frac{1}{6} \ln 2\right)=-\frac{1}{6} \ln 2 \end{aligned}

$3:\left\{\begin{array}{l}1: \text { denominator of } f^{\prime}(x) \\ 1: f^{\prime}(x) \\ 1: \text { answer }\end{array}\right.$ $3:\left\{\begin{array}{r}1: \text { partial fraction } \\ \text { decomposition } \\ 1: \text { antiderivatives } \\ 1: \text { answer }\end{array}\right.$

All question bank results loaded

Covered syllabus points

Question 1