Primitivas de funciones racionales
Las primitivas de las funciones racionales se deducen por las de su descomposición en elementos simples , por lo tanto de las siguientes fórmulas:
(Suponemos un ≠ 0. )
∫(aX+B)noDX=1(no+1)a(aX+B)no+1+VS{\ Displaystyle \ int (ax + b) ^ {n} \, \ mathrm {d} x = {\ frac {1} {(n + 1) a}} (ax + b) ^ {n + 1} + VS}![{\ Displaystyle \ int (ax + b) ^ {n} \, \ mathrm {d} x = {\ frac {1} {(n + 1) a}} (ax + b) ^ {n + 1} + VS}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afbd714fd5f1ca0137c0c5e7596bcf1a2c97d329)
para cualquier
entero relativo n distinto de -1 ( fórmula de cuadratura de Cavalieri
(en) )
∫1aX+BDX=1aen|aX+B|+VS{\ Displaystyle \ int {\ frac {1} {ax + b}} \, \ mathrm {d} x = {\ frac {1} {a}} \ ln | ax + b | + C}
∫1aX2+BX+vsDX={2-(B2-4avs)arctan2aX+B-(B2-4avs)+VS Si B2-4avs<0-22aX+B+VS Si B2-4avs=01B2-4avsen|2aX+B-B2-4avs2aX+B+B2-4avs|+VS={-2B2-4avsArtanh2aX+BB2-4avs+VS Si |2aX+B|<B2-4avs-2B2-4avsarcot2aX+BB2-4avs+VS si no Si B2-4avs>0{\ Displaystyle \ int {\ frac {1} {ax ^ {2} + bx + c}} \, \ mathrm {d} x = \ left \ {{\ begin {array} {lll} \ displaystyle {\ frac {2} {\ sqrt {- (b ^ {2} -4ac)}}} \ operatorname {arctan} {\ frac {2ax + b} {\ sqrt {- (b ^ {2} -4ac)}}} + C & {\ text {si}} & b ^ {2} -4ac <0 \\ [18pt] \ displaystyle {\ frac {-2} {2ax + b}} + C & {\ text {si}} & b ^ {2} -4ac = 0 \\ [6pt] \ displaystyle {\ frac {1} {\ sqrt {b ^ {2} -4ac}}} \ ln \ left | {\ frac {2ax + b - {\ sqrt {b ^ {2} -4ac}}} {2ax + b + {\ sqrt {b ^ {2} -4ac}}}} \ right | + C = \ left \ {{\ begin {array} {ll} - {\ frac {2} {\ sqrt {b ^ {2} -4ac}}} \ operatorname {artanh} {\ frac {2ax + b} {\ sqrt {b ^ {2} -4ac}} } + C & {\ text {si}} | 2ax + b | <{\ sqrt {b ^ {2} -4ac}} \\ - {\ frac {2} {\ sqrt {b ^ {2} -4ac }}} \ operatorname {arcoth} {\ frac {2ax + b} {\ sqrt {b ^ {2} -4ac}}} + C & {\ text {de lo contrario}} \ end {array}} \ right. & {\ text {si}} & b ^ {2} -4ac> 0 \ end {array}} \ right.}
∫XaX2+BX+vsDX=12aen|aX2+BX+vs|-B2a∫1aX2+BX+vsDX{\ Displaystyle \ int {\ frac {x} {ax ^ {2} + bx + c}} \, \ mathrm {d} x = {\ frac {1} {2a}} \ ln | ax ^ {2} + bx + c | - {\ frac {b} {2a}} \ int {\ frac {1} {ax ^ {2} + bx + c}} \, \ mathrm {d} x}![{\ Displaystyle \ int {\ frac {x} {ax ^ {2} + bx + c}} \, \ mathrm {d} x = {\ frac {1} {2a}} \ ln | ax ^ {2} + bx + c | - {\ frac {b} {2a}} \ int {\ frac {1} {ax ^ {2} + bx + c}} \, \ mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db80b6ec2974385b19078743afd1a0c9eec38744)
Para cualquier número entero n ≥ 2 :
- ∫1(aX2+BX+vs)noDX=-2aX+B(no-1)(B2-4avs)(aX2+BX+vs)no-1-2(2no-3)a(no-1)(B2-4avs)∫1(aX2+BX+vs)no-1DX{\ Displaystyle {\ begin {alineado} \ int {\ frac {1} {(ax ^ {2} + bx + c) ^ {n}}} \, \ mathrm {d} x = - {\ frac {2ax + b} {(n-1) (b ^ {2} -4ac) (ax ^ {2} + bx + c) ^ {n-1}}} - {\ frac {2 (2n-3) a} {(n-1) (b ^ {2} -4ac)}} \ int {\ frac {1} {(ax ^ {2} + bx + c) ^ {n-1}}} \, \ mathrm { d} x \ end {alineado}}}
![{\ Displaystyle {\ begin {alineado} \ int {\ frac {1} {(ax ^ {2} + bx + c) ^ {n}}} \, \ mathrm {d} x = - {\ frac {2ax + b} {(n-1) (b ^ {2} -4ac) (ax ^ {2} + bx + c) ^ {n-1}}} - {\ frac {2 (2n-3) a} {(n-1) (b ^ {2} -4ac)}} \ int {\ frac {1} {(ax ^ {2} + bx + c) ^ {n-1}}} \, \ mathrm { d} x \ end {alineado}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca79524f17fd85d919ca40177e89163cc354432f)
- ∫X(aX2+BX+vs)noDX=BX+2vs(no-1)(B2-4avs)(aX2+BX+vs)no-1+(2no-3)B(no-1)(B2-4avs)∫1(aX2+BX+vs)no-1DX{\ Displaystyle \ int {\ frac {x} {(ax ^ {2} + bx + c) ^ {n}}} \, \ mathrm {d} x = {\ frac {bx + 2c} {(n- 1) (b ^ {2} -4ac) (ax ^ {2} + bx + c) ^ {n-1}}} + {\ frac {(2n-3) b} {(n-1) (b ^ {2} -4ac)}} \ int {\ frac {1} {(ax ^ {2} + bx + c) ^ {n-1}}} \, \ mathrm {d} x}
![{\ Displaystyle \ int {\ frac {x} {(ax ^ {2} + bx + c) ^ {n}}} \, \ mathrm {d} x = {\ frac {bx + 2c} {(n- 1) (b ^ {2} -4ac) (ax ^ {2} + bx + c) ^ {n-1}}} + {\ frac {(2n-3) b} {(n-1) (b ^ {2} -4ac)}} \ int {\ frac {1} {(ax ^ {2} + bx + c) ^ {n-1}}} \, \ mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd2ed8a561bba7375aa71ec9531d58e2c18126a)
Artículo relacionado
Tabla de primitivas
<img src="https://fr.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">