[tex]2)\int_{0}^{1}(2x - 3) \: dx[/tex]
[tex] \int(2x - 3) \: dx = \int2x \: dx - \int3 \: dx[/tex]
[tex] = 2 \int x \: dx - 3 \int1 \: dx [/tex]
[tex] = 2 \times \frac{ {x}^{1 + 1} }{1 + 1} - 3x = \frac{2 {x}^{2} }{2} - 3x = {x}^{2} - 3x + C[/tex]
[tex]\int_{0}^{1}(2x - 3) \: dx = ( {x}^{2} - 3x) |_{0}^{1}[/tex]
[tex] = {1}^{2} - 3 \times 1 - ( {0}^{2} - 3 \times 0)[/tex]
[tex] = 1 - 3 - 0 = - 2[/tex]
[tex]3) \int_{ - 1}^{1}3 {x}^{3} \: dx[/tex]
[tex] \int3 {x}^{3} \: dx = 3 \int {x}^{3} \: dx = 3 \times \frac{ {x}^{3 + 1} }{3 + 1}
= \frac{3 {x}^{4} }{4} + C[/tex]
[tex]
\int_{ - 1}^{1}3 {x}^{3}\:dx = \frac{3 {x}^{4} }{4}
| _{ - 1}^{1} = \frac{3 \times {1}^{4} }{4} - \frac{3 \times {( - 1)}^{4} }{4} [/tex]
[tex] = \frac{3 \times 1}{4} - \frac{3 \times 1}{4} = \frac{3}{4} - \frac{3}{4} = 0[/tex]
