a: \(\int\left(2x-1\right)\left(x+2\right)dx=\int\left(2x^2+3x-2\right)dx\)
\(=2\cdot\dfrac{1}{3}x^3+3\cdot\dfrac{1}{2}x^2-2x=\dfrac{2}{3}x^3+\dfrac{3}{2}x^2-2x\)
\(F=\int_0^1\left(2x-1\right)\left(x+2\right)dx\)
\(=\left(\dfrac{2}{3}\cdot1^3+\dfrac{3}{2}\cdot1^2-2\cdot1\right)-\left(\dfrac{2}{3}\cdot0^3+\dfrac{3}{2}\cdot0^2-2\cdot0\right)\)
\(=\dfrac{2}{3}+\dfrac{3}{2}-2=\dfrac{13}{6}-2=\dfrac{1}{6}\)
b: \(\int x\left(2x-1\right)^2dx=\int\left(4x^3-4x^2+x\right)dx\)
\(=4\cdot\dfrac{1}{4}x^4-4\cdot\dfrac{1}{3}x^3+\dfrac{1}{2}x^2=x^4-\dfrac{4}{3}x^3+\dfrac{1}{2}x^2\)
\(F=\int_1^2x\left(2x-1\right)^2dx\)
\(=\left(2^4-\dfrac{4}{3}\cdot2^3+\dfrac{1}{2}\cdot2^2\right)-\left(1^4-\dfrac{4}{3}\cdot1^3+\dfrac{1}{2}\cdot1^2\right)\)
\(=\left(16-\dfrac{32}{3}+2\right)-\left(1-\dfrac{4}{3}+\dfrac{1}{2}\right)\)
\(=\dfrac{43}{6}\)
