Statement (a) Decompose 5x−4(x−2)(x+1)\dfrac{5x-4}{(x-2)(x+1)}(x−2)(x+1)5x−4 into partial fractions. (b) Compute ∫355x−4(x−2)(x+1) dx\displaystyle\int_3^5 \frac{5x-4}{(x-2)(x+1)}\,dx∫35(x−2)(x+1)5x−4dx. Solution (a) 5x−4(x−2)(x+1)=Ax−2+Bx+1\dfrac{5x-4}{(x-2)(x+1)}=\dfrac{A}{x-2}+\dfrac{B}{x+1}(x−2)(x+1)5x−4=x−2A+x+1B, i.e. 5x−4=A(x+1)+B(x−2)5x-4=A(x+1)+B(x-2)5x−4=A(x+1)+B(x−2). For x=2x=2x=2: 6=3A⇒A=26=3A\Rightarrow A=26=3A⇒A=2; for x=−1x=-1x=−1: −9=−3B⇒B=3-9=-3B\Rightarrow B=3−9=−3B⇒B=3. (b) ∫35(2x−2+3x+1)dx=[2ln∣x−2∣+3ln∣x+1∣]35\displaystyle\int_3^5\Big(\frac{2}{x-2}+\frac{3}{x+1}\Big)dx=\big[2\ln|x-2|+3\ln|x+1|\big]_3^5∫35(x−22+x+13)dx=[2ln∣x−2∣+3ln∣x+1∣]35 =(2ln3+3ln6)−(0+3ln4)=2ln3+3ln6−3ln4≈3.414=(2\ln3+3\ln6)-(0+3\ln4)=2\ln3+3\ln6-3\ln4\approx 3{.}414=(2ln3+3ln6)−(0+3ln4)=2ln3+3ln6−3ln4≈3.414. ∫355x−4(x−2)(x+1) dx≈3.414\boxed{\int_3^5\frac{5x-4}{(x-2)(x+1)}\,dx\approx 3{.}414}∫35(x−2)(x+1)5x−4dx≈3.414