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## Homework Statement

Are both methods and final results correct?

[itex]\int tan^{5}xdx[/itex]

Method 1:

[itex]\int tan^{3}xtan^{2}xdx[/itex]

[itex]\int tanx^{3} (sec^{2}x - 1) dx[/itex]

[itex]\int tan^{3}xsec^{2}x dx - \int tanxtan^{2}x dx[/itex]

[itex]\int tan^{3}xsec^{2}x dx - \int tanxsec^{2}x dx + \int tanx[/itex]

[itex]\int u^{3}du - \int u du - \int \frac{1}{u} du[/itex]

[itex]\frac{1}{4}tan^{4}x - \frac{1}{2}tanx - ln|cosx| + C[/itex]

Method 2:

[itex]\int tan^{5}x dx[/itex]

[itex]\int tan^{4}tanx[/itex]

[itex]\int (sec^{2}x - 1)^{2}tanxdx [/itex]

[itex]\int (sec^{4}x - 2sec^{2}x + 1)tanx dx [/itex]

[itex]\int (u^{2} - 2u +1) du[/itex]

[itex]\frac{1}{3}tan^{3}x - tan^{2}x + tanx + C[/itex]