21-25 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 23. \int_(-2)^(0)\left(x^(2)+x\right) d x

Question

21-25 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.

23.
`\int_(-2)^(0)\left(x^(2)+x\right) d x`

Answer 1

`\displaystyle \int^0_(-2) (x^2 +x )~dx\\\\\\=\displaystyle \int^0_(-2) x^2~ dx +\displaystyle \int^0_(-2)x ~ dx\\\\\\=\frac 13 \left[x^3 \right]^(0)_(-2)+ \frac 12 \left[ x^2\right]^(0)_(-2)\\\\\\=\frac 13 (0+8) + \frac 12 (0-4)\\ \\\\=\frac 83 - \frac 42\\\\\\=\frac 83 - 2\\\\\\=\frac 23`