[matemáticas] \ begin {align} I & = \ int x \ tan x \, \ mathrm dx \\ & = – \ int \ dfrac {ix (e ^ {ix} -e ^ {- ix})} {e ^ {ix} + e ^ {- ix}} \ mathrm dx \\ & = – \ int \ dfrac {ix (e ^ {2ix} -1)} {e ^ {2ix} +1} \ mathrm dx \\\ hline & \ text {let} u = e ^ {2ix} +1 \ implica \ mathrm du = 2ie ^ {2ix} \ mathrm dx \\ & \ qquad \ qquad \ qquad \ implica \ dfrac {\ mathrm du} {2ie ^ {2ix}} = \ mathrm dx \\ & \ qquad \ qquad \ qquad \ implica \ dfrac {\ mathrm du} {2i (u-1)} = \ mathrm dx \\ & 2ix = \ ln (u-1) \ implica ix = \ dfrac 12 \ ln (u-1) \\\ hline & = – \ int \ dfrac {\ dfrac12 (u-2) \ ln (u-1)} {2iu (u-1)} \ mathrm du \\ & = \ dfrac i4 \ int \ left [\ dfrac {u-2} {u (u-1)} \ right] \ ln (u-1) \ mathrm du \\ & = \ dfrac i4 \ int \ left [\ dfrac {2 (u-1) -u} {u (u-1)} \ right] \ ln (u-1) \ mathrm du \\ & = \ dfrac i4 \ int \ left [\ dfrac {2 \ ln (u-1)} u- \ dfrac {\ ln (u-1)} {u-1} \ right] \ mathrm du \\ & = \ dfrac i2 \ int \ dfrac {\ ln (- 1) + \ ln (1-u)} u \ mathrm du- \ dfrac i4 \ int \ dfrac {\ ln (u-1)} {u-1} \ mathrm du \\ & = \ dfrac i2 \ int \ dfrac {\ ln (-1)} u \ mathrm du + \ dfrac i2 \ int \ dfrac {\ ln (1-u)} u \ mathrm du- \ dfrac i4 \ int \ dfrac {\ ln (u-1)} {u-1} \ mathrm du \\ & = \ dfrac i2 \ ln (-1) \ ln u- \ dfrac i2 \ int- \ dfrac {\ ln (1-u)} u \ mathrm du- \ dfrac i8 \ ln ^ 2 (u-1) + C \\ & = \ dfrac i2 \ ln (-1) \ ln u- \ dfrac i2 \ mathrm {Li} _2 (u) – \ dfrac i8 \ ln ^ 2 (u-1) + C \\ & = \ dfrac i2 \ ln (-1) \ ln (e ^ {2ix} +1) – \ dfrac i2 \ mathrm {Li} _2 (e ^ {2ix} +1) – \ dfrac i8 (2ix ) ^ 2 + C \\ & = \ dfrac i2 \ ln (-1) \ ln (e ^ {2ix} +1) – \ dfrac i2 \ mathrm {Li} _2 (e ^ {2ix} +1) + \ dfrac {ix ^ 2} 2 + C \\ & = \ bbox [2pt, borde: # 10f 2pt sólido] {\ bbox [# 7FFFD0] {\ dfrac i2 \ bigg [\ ln (-1) \ ln (e ^ {2ix} +1) – \ mathrm {Li} _2 (e ^ {2ix} +1) + x ^ 2 \ bigg] + C}} \ end {align} \ tag * {} [/ math]
- Resultado utilizado: Dilogaritmo , o Función de Spence [matemáticas] \ displaystyle \ int- \ dfrac {\ ln (1-x)} x \ mathrm dx = \ mathrm {Li} _2 (x) \ tag * {} [/ math]