[tex]1) \int(1 - x + {x}^{2} ) \: dx[/tex]
[tex] = \int1 \: dx - \int x \: dx + \int {x}^{2} \: dx[/tex]
[tex] = x - \frac{ {x}^{1 + 1} }{1 + 1} + \frac{ {x}^{2 + 1} }{2 + 1} [/tex]
[tex] = x - \frac{ {x}^{2} }{2} + \frac{ {x}^{3} }{3} + C[/tex]
[tex]2) \int(3 + {x}^{3} ) \: dx[/tex]
[tex] = \int3 \: dx + \int {x}^{3} \: dx[/tex]
[tex] = 3 \int1 \: dx + \frac{ {x}^{3 + 1} }{3 + 1} [/tex]
[tex] = 3x + \frac{ {x}^{4} }{4} + C[/tex]
[tex]3) \int(5 {x}^{2} - 4x) \: dx[/tex]
[tex] = \int5 {x}^{2} \: dx - \int4x \: dx[/tex]
[tex] = 5 \int {x}^{2} - 4 \int x \: dx[/tex]
[tex] = 5 \times \frac{ {x}^{2 + 1} }{2 + 1} - 4 \times \frac{ {x}^{1 + 1} }{1 + 1} [/tex]
[tex] = 5 \times \frac{ {x}^{3} }{3} - 4 \times \frac{ {x}^{2} }{2} [/tex]
[tex] = \frac{5 {x}^{3} }{3} - 2 {x}^{2} + C[/tex]
[tex]4) \int {(x - 3)}^{2} \: dx[/tex]
[tex] = \int( {x}^{2} - 6x + 9) \: dx[/tex]
[tex] = \int {x}^{2} \: dx - \int6x \: dx + \int9 \: dx[/tex]
[tex] = \frac{ {x}^{2 + 1} }{2 + 1} - 6 \int x \: dx + 9 \int1 \: dx[/tex]
[tex] = \frac{ {x}^{3} }{3} - 6 \times \frac{ {x}^{1 + 1} }{1 + 1} + 9x[/tex]
[tex] = \frac{ {x}^{3} }{3} - 6 \times \frac{ {x}^{2} }{2} + 9x[/tex]
[tex] = \frac{ {x}^{3} }{3} - 3 {x}^{2} + 9x + C[/tex]
[tex]5) \int \frac{1}{ {x}^{3} } \: dx = \int {x}^{ - 3} \: dx = \frac{ {x}^{ - 3 + 1} }{ - 3 + 1} = \frac{ {x}^{ - 2} }{ - 2} = - \frac{ {x}^{ - 2} }{2} + C[/tex]
