¿Cuál es la integración de dx / (x ^ 3 +3)?

Answer 1

Deje [math] T = \ displaystyle \ int \ frac {1} {x ^ 3 + 3} \, \ mathrm dx [/ math].

Vamos a tener muchas raíces cúbicas volando, así que escribamos [math] a = \ sqrt [3] {3}. [/ Math]

[matemáticas] \ begin {align} \ por lo tanto T & = \ int \ frac {1} {x ^ 3 + a ^ 3} \\ & = \ int \ frac {1} {\ left (x + a \ right) \ left (x ^ 2 – ax + a ^ 2 \ right)} \, \ mathrm dx \\ & = \ int \ left (\ frac {1} {3a ^ 2 \ left (x + a \ right)} – \ frac {x – 2a} {3a ^ 2 \ left (x ^ 2 – ax + a ^ 2 \ right)} \ right) \, \ mathrm dx \\ & = \ frac {1} {3a ^ 2} \ int \ frac {1} {x + a} \, \ mathrm dx – \ frac {1} {3a ^ 2} \ int \ frac {x – {\ textstyle \ frac {1} {2}} a – {\ textstyle \ frac {3} {2}} a} {x ^ 2 – ax + a ^ 2} \, \ mathrm dx \\ & = \ frac {1} {3a ^ 2} \ log \ left \ lvert x + a \ right \ rvert – \ frac {1} {6a ^ 2} \ int \ frac {2x – a} {x ^ 2 – ax + a ^ 2} \, \ mathrm dx + \ frac {1} {2a} \ int \ frac {1} {x ^ 2 – ax + a ^ 2} \, \ mathrm dx \\ & = \ frac {1} {6a ^ 2} \ log \ left \ lvert x + a \ right \ rvert ^ 2 – \ frac {1} {6a ^ 2} \ log \ left \ lvert x ^ 2 – ax + a ^ 2 \ right \ rvert + \ frac {2} {a} \ int \ frac {1} {\ left (2x – a \ right) ^ 2 + 3a ^ 2} \, \ mathrm dx \ end {align} [/ math]

Sea [math] u = \ dfrac {2x – a} {\ sqrt {3} a} \ \ por lo tanto 2 \, \ mathrm dx = \ sqrt {3} a \, \ mathrm du [/ math].

[matemáticas] \ begin {align} \ por lo tanto \ int \ frac {1} {\ left (2x – a \ right) ^ 2 + 3a ^ 2} \, \ mathrm dx & = \ frac {\ sqrt {3} a } {2} \ int \ frac {1} {3a ^ 2u ^ 2 + 3a ^ 2} \, \ mathrm du \\ & = \ frac {1} {2 \ sqrt {3} a} \ int \ frac { 1} {u ^ 2 + 1} \, \ mathrm du \\ & = \ frac {1} {2 \ sqrt {3} a} \ arctan {u} \\ & = \ frac {1} {2 \ sqrt {3} a} \ arctan \ left (\ frac {2x – a} {\ sqrt {3} a} \ right) \ end {align} [/ math]

[matemáticas] \ begin {align} \ por lo tanto T & = \ frac {1} {6a ^ 2} \ log \ left \ lvert \ frac {x ^ 2 + 2ax + a ^ 2} {x ^ 2 – ax + a ^ 2} \ right \ rvert + \ frac {1} {\ sqrt {3} a ^ 2} \ arctan \ left (\ frac {2x – a} {\ sqrt {3} a} \ right) \\ & = \ boxed {\ frac {1} {6 \ sqrt [3] {9}} \ log \ left \ lvert \ frac {x ^ 2 + 2 \ sqrt [3] {3} x + \ sqrt [3] {9 }} {x ^ 2 – \ sqrt [3] {3} x + \ sqrt [3] {9}} \ right \ rvert + \ frac {1} {\ sqrt {3} \ sqrt [3] {9} } \ arctan \ left (\ frac {2x – \ sqrt [3] {3}} {\ sqrt {3} \ sqrt [3] {3}} \ right) + C} \ end {align} [/ math]

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Letra pequeña: • cuando uso [math] \ log [/ math] esto denota un logaritmo natural (base [math] e [/ math] ); si se pretende cualquier otra base, se mostrará como un subíndice • Generalmente omito las constantes de integración de integrales indefinidas hasta que muestre un resultado final •

Answer 2

[matemáticas] \ displaystyle \ int \ frac {dx} {x ^ 3 + 3} dx = [/ matemáticas]

[matemáticas] \ displaystyle 3 ^ {\ frac {5} {6}} \ left (- \ frac {3 ^ {\ frac {5} {6}} d} {27} \ ln (| x + \ sqrt [3 ] {3} |) + \ frac {3 ^ {\ frac {5} {6}} d} {54} \ ln \ left (| x ^ 2- \ sqrt [3] {3} x + 3 ^ { \ frac {2} {3}} \ right | \ right) + \ frac {\ sqrt [3] {3} d} {9} atan \ left (\ sqrt [6] {3} \ left (\ frac { 2x} {3} – \ frac {\ sqrt [3] {3}} {3} \ right) \ right) [/ math]