Answer:
[tex]=2x^2-x+5\ln \left|x-2\right|+\frac{1}{3}\left(x-2\right)^3-6+C[/tex]
Step-by-step explanation:
[tex]\int \frac{x^3-2x^2+3x-1}{x-2}dx[/tex]
[tex]\frac{x^3-2x^2+3x-1}{x-2}\\[/tex]
[tex]=\frac{x^3}{x-2}-\frac{2x^2}{x-2}+\frac{3x}{x-2}-\frac{1}{x-2}[/tex]
[tex]=\int \frac{x^3}{x-2}dx-\int \frac{2x^2}{x-2}dx+\int \frac{3x}{x-2}dx-\int \frac{1}{x-2}dx[/tex]
[tex]\int \frac{x^3}{x-2}dx[/tex]
[tex]=\int \frac{\left(u+2\right)^3}{u}du; u=x-2[/tex]
[tex]\frac{\left(u+2\right)^3}{u}[/tex]
[tex]\left(u+2\right)^3[/tex]
[tex]=u^3+3u^2\cdot \:2+3u\cdot \:2^2+2^3[/tex]
[tex]=u^3+6u^2+12u+8[/tex]
[tex]=\frac{u^3+6u^2+12u+8}{u}[/tex]
[tex]\frac{u^3+6u^2+12u+8}{u}=\frac{u^3}{u}+\frac{6u^2}{u}+\frac{12u}{u}+\frac{8}{u}[/tex]
[tex]=u^2+\frac{6u^2}{u}+\frac{12u}{u}+\frac{8}{u}[/tex]
[tex]=u^2+6u+\frac{12u}{u}+\frac{8}{u}[/tex]
[tex]=u^2+6u+12+\frac{8}{u}[/tex]
[tex]=\int \:u^2+6u+12+\frac{8}{u}du[/tex]
[tex]=\int \:u^2du+\int \:6udu+\int \:12du+\int \frac{8}{u}du[/tex]
[tex]\int \:u^2du[/tex]
[tex]=\frac{u^{2+1}}{2+1}[/tex]
[tex]=\frac{u^3}{3}[/tex]
[tex]\int \:6udu[/tex]
[tex]=6\cdot \int \:udu[/tex]
[tex]=6\cdot \frac{u^{1+1}}{1+1}[/tex]
[tex]=3u^2[/tex]
[tex]\int \:12du[/tex]
[tex]=12u[/tex]
[tex]\int \frac{8}{u}du[/tex]
[tex]=8\cdot \int \frac{1}{u}du[/tex]
[tex]=8\ln \left|u\right|[/tex]
[tex]=\frac{u^3}{3}+3u^2+12u+8\ln \left|u\right|[/tex]
[tex]=\frac{\left(x-2\right)^3}{3}+3\left(x-2\right)^2+12\left(x-2\right)+8\ln \left|x-2\right|[/tex]
[tex]=3x^2+8\ln \left|x-2\right|+\frac{1}{3}\left(x-2\right)^3-12[/tex]
[tex]\int \frac{2x^2}{x-2}dx[/tex]
[tex]=2\cdot \int \frac{x^2}{x-2}dx[/tex]
[tex]=2\cdot \int \frac{\left(u+2\right)^2}{u}du[/tex]
[tex]\left(u+2\right)^2[/tex]
[tex]=u^2+4u+4[/tex]
[tex]=\frac{u^2+4u+4}{u}[/tex]
[tex]\frac{u^2+4u+4}{u}=\frac{u^2}{u}+\frac{4u}{u}+\frac{4}{u}[/tex]
[tex]=u+\frac{4u}{u}+\frac{4}{u}[/tex]
[tex]=2\cdot \int \:u+4+\frac{4}{u}du[/tex]
[tex]=2\left(\int \:udu+\int \:4du+\int \frac{4}{u}du\right)[/tex]
[tex]\int \:udu[/tex]
[tex]=\frac{u^{1+1}}{1+1}[/tex]
[tex]=\frac{u^2}{2}[/tex]
[tex]\int \:4du[/tex]
[tex]=4u[/tex]
[tex]\int \frac{4}{u}du[/tex]
[tex]=4\cdot \int \frac{1}{u}du[/tex]
[tex]=4\ln \left|u\right|[/tex]
[tex]=2\left(\frac{u^2}{2}+4u+4\ln \left|u\right|\right)[/tex]
[tex]=2\left(\frac{\left(x-2\right)^2}{2}+4\left(x-2\right)+4\ln \left|x-2\right|\right)[/tex]
[tex]2\left(\frac{\left(x-2\right)^2}{2}+4\left(x-2\right)+4\ln \left|x-2\right|\right)[/tex]
[tex]=2\cdot \frac{\left(x-2\right)^2}{2}+2\cdot \:4\left(x-2\right)+2\cdot \:4\ln \left|x-2\right|[/tex]
[tex]=x^2+4x+8\ln \left|x-2\right|-12[/tex]
[tex]\int \frac{3x}{x-2}dx[/tex]
[tex]=3\cdot \int \frac{x}{x-2}dx[/tex]
[tex]=3\cdot \int \frac{u+2}{u}du[/tex]
[tex]=3\cdot \int \:1+\frac{2}{u}du[/tex]
[tex]=3\left(\int \:1du+\int \frac{2}{u}du\right)[/tex]
[tex]\int \:1du[/tex]
[tex]=u[/tex]
[tex]\int \frac{2}{u}du[/tex]
[tex]=2\cdot \int \frac{1}{u}du[/tex]
[tex]=2\ln \left|u\right|[/tex]
[tex]=3\left(u+2\ln \left|u\right|\right)[/tex]
[tex]=3\left(x-2+2\ln \left|x-2\right|\right)[/tex]
[tex]\int \frac{1}{x-2}dx[/tex]
[tex]=\int \frac{1}{u}du[/tex]
[tex]=\ln \left|u\right|[/tex]
[tex]=\ln \left|x-2\right|[/tex]
[tex]3x^2+8\ln \left|x-2\right|+\frac{1}{3}\left(x-2\right)^3-12-\left(x^2+4x+8\ln \left|x-2\right|-12\right)\\+3\left(x-2+2\ln \left|x-2\right|\right)-\ln \left|x-2\right|[/tex]
[tex]=2x^2-x+5\ln \left|x-2\right|+\frac{1}{3}\left(x-2\right)^3-6[/tex]
[tex]=2x^2-x+5\ln \left|x-2\right|+\frac{1}{3}\left(x-2\right)^3-6+C[/tex]
