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[aifC]

1.3 [tex]\int \left(12x^{2}-x+\frac{2}{x} \right)dx = 12\int x^{2}dx - \int xdx + 2\int \frac{1}{x}dx = 2ln(|x|)+4x^{3}-\frac{x^{2}}{2} +C;[/tex]

2.3 [tex]\int \frac{6x^{5}+3x^{2}-2x}{x^{3}}dx = \int \frac{6x^{4}+3x-2x}{x^{2}}dx = \int \left(6x^{2} + \frac{3}{x} - \frac{2}{x^{2}} \right)dx = 6\int x^{2}dx + 3\int \frac{1}{x}dx -2\int \frac{1}{x^{2}}dx = 3ln(|x|)+2x^{3}+\frac{2}{x} + C;[/tex]

3.3 [tex]\int \frac{5-3x+3x^{3}}{x^{5}}dx = \int \left(\frac{3}{x^{2}}-\frac{3}{x^{4}}+\frac{5}{x^{5}} \right)dx = 3\int \frac{1}{x^{2}}dx - 3\int \frac{1}{x^{4}}dx+5\int \frac{1}{x^{5}}dx = -\frac{12x^{3}-4x+5}{4x^{4}}+C;[/tex]

4.3 [tex]\int(4ay^{5}-5a^{2}y)dy = 4a\int y^{5}dy - 5a^{2}\int ydy = \frac{ay^{2}(4y^{4}-15a)}{6} + C;[/tex]

5.3 [tex]\int\frac{4x^{2}\sqrt[3]{x^{2}}-5x\sqrt{x}}{x}dx = \int 4x^{\frac{5}{3}}dx - \int 5\sqrt{x}dx = \frac{3}{2}x^{\frac{8}{3}}-\frac{10}{3}x^{\frac{3}{2}}+C;[/tex]

6.3 [tex]\int sin(3-x)dx = \int -sin(x-3); u=x-3 \rightarrow dx=du \Rightarrow \int sin(u)du = cos(x-3) + C;[/tex]

7.3 [tex]\int sin^{2}(2x-3)dx; u=2x-3 \rightarrow dx=\frac{1}{2}du \Rightarrow \frac{1}{2}\int sin^{2}(u)du = \frac{1}{2}\int 1du - \frac{cos(u)sin(u)}{2} = -\frac{sin(4x-6)-4x}{8}+C;[/tex]

8.3 [tex]\int \frac{1}{sin^{2}(x-3)}dx; u=x-3 \rightarrow dx=du \Rightarrow \int csc^{2}du = -cotg(x-3)+C[/tex]

9.3 [tex]\int \frac{2}{x^{2}-1}dx = 2\int \frac{1}{x^{2}-1}dx = \int \left(\frac{1}{2(x-1)}-\frac{1}{2(x+1)} \right)dx = \frac{1}{2}\int \frac{1}{x-1}dx - \frac{1}{2}\int \frac{1}{x+1}dx; u=x+1 \rightarrow dx=du \Rightarrow ln(|x-1|)-ln(|x+1|)+C;[/tex]

10.3 [tex]\int \frac{2}{x^{2}+4}dx; u=\frac{x}{2} \rightarrow dx=2du \Rightarrow \int \frac{1}{u^{2}+1}du = arctg \left(\frac{x}{2} \right)+C;[/tex]

11.3 [tex]\int \frac{1}{\sqrt{x^{2}+5}}dx; u = \frac{x}{\sqrt{5}} \rightarrow dx = \sqrt{5}du \Rightarrow \int \frac{1}{\sqrt{u^{2}+1}}du; u=tg(v) \rightarrow v=arctg(u), du=sec^{2}(v)dv \Rightarrow \int \frac{sec^{2}(v)}{\sqrt{tg^{2}(v)+1}}dv = \int sec(v)dv = \int \frac{sec(v)tg(v)+sec^{2}(v)}{tg(v)+sec(v)}dv;
w = tg(v)+sec(v) \rightarrow dv = \frac{1}{sec(v)tg(v)+sec^{2}(v)}dw \Rightarrow \int \frac{1}{w}dw = arcsinh \left(\frac{x}{\sqrt{5}} \right)+C;[/tex]

12.3 [tex]\int \frac{8}{\sqrt{5-x^{2}}}dx; u=\frac{x}{\sqrt{5}} \rightarrow dx=\sqrt{5}du \Rightarrow 8 \int \frac{1}{\sqrt{1-u^{2}}}du =
8arcsin \left(\frac{x}{\sqrt{5}} \right)+C;[/tex]