Abróchese el cinturón de seguridad: este será un viaje ligeramente lleno de baches. Tenga en cuenta que en esta respuesta uso [math] \ log [/ math] para el logaritmo natural, es decir , con base [math] e [/ math].
[matemáticas] \ begin {align} \ text {Set} T & = \ int \ frac {1} {\ sqrt {x ^ 2 + 7x} +3} \, \ mathrm dx \\ & = \ int \ frac { \ sqrt {x ^ 2 + 7x} – 3} {\ left (\ sqrt {x ^ 2 + 7x} + 3 \ right) \ left (\ sqrt {x ^ 2 + 7x} – 3 \ right)} \, \ mathrm dx \\ & = \ int \ frac {\ sqrt {x ^ 2 + 7x}} {x ^ 2 + 7x – 9} \, \ mathrm dx – 3 \ int \ frac {1} {x ^ 2 + 7x – 9} \, \ mathrm dx \ end {align} [/ math]
Establezca [math] U = \ displaystyle \ int \ frac {\ sqrt {x ^ 2 + 7x}} {x ^ 2 + 7x – 9} \, \ mathrm dx [/ math]
Establezca [math] V = \ displaystyle \ int \ frac {1} {x ^ 2 + 7x – 9} \, \ mathrm dx [/ math]
- Cómo calcular [math] \ displaystyle \ sum_ {n = 1} ^ {\ infty} \ frac {2n-1} {2 ^ n} [/ math]
- ¿Cuál es el valor de x donde x ^ 12 = 2?
- ¿Cuál es el supremum de f (x, y) con x, y en [0,2] donde f (x, y) = (x ^ 4-y ^ 4-4 (x ^ 3-y ^ 3) +4 (x ^ 2-y ^ 2)) / 4 (x ^ 2-y ^ 2) si x = / = y y f (x, y) = 0 si x = y?
- ¿Por qué los números arábigos son ‘árabes’, si fue fundado por un persa?
- Pregunta práctica: ¿Cómo encuentro el cuadrado de cualquier número?
Entonces [matemáticas] T = U – 3V [/ matemáticas]
[matemáticas] \ begin {align} U & = \ int \ frac {\ sqrt {x ^ 2 + 7x}} {x ^ 2 + 7x – 9} \, \ mathrm dx \ end {align} [/ math]
Establezca [matemática] u = x + \ frac {7} {2} \ \ por lo tanto \ matemática du = \ matemática dx [/ matemática]
[matemáticas] \ begin {align} \ por lo tanto U & = \ int \ frac {\ sqrt {u ^ 2 – \ frac {49} {4}}} {u ^ 2 – \ frac {85} {4}} \ , \ mathrm du \ end {align} [/ math]
Establezca [matemáticas] u = \ frac {7} {2 \ cos {v}} \ \ por lo tanto \ mathrm du = \ frac {7 \ sin {v}} {2 \ cos ^ 2 {v}} \, \ mathrm dv [/ math]
[matemáticas] \ begin {align} \ por lo tanto U & = \ int \ frac {7 \ sin {v} \, \ sqrt {\ frac {49} {4 \ cos ^ 2 {v}} – \ frac {49} {4}}} {2 \ cos ^ 2 {v} \ left (\ frac {49} {4 \ cos ^ 2 {v}} – \ frac {85} {4} \ right)} \, \ mathrm dv \\ & = 49 \ int \ frac {\ sin {v} \, \ sqrt {\ frac {1} {\ cos ^ 2 {v}} – 1}} {49 – 85 \ cos ^ 2 {v}} \, \ mathrm dv \\ & = 49 \ int \ frac {\ sin {v} \, \ sqrt {\ frac {1} {\ cos ^ 2 {v}} – 1}} {\ sin ^ 2 {v } \ left (\ frac {49} {\ sin ^ 2 {v}} – \ frac {85} {\ tan ^ 2 {v}} \ right)} \, \ mathrm dv \ end {align} [/ math ]
[matemáticas] \ frac {1} {\ cos ^ 2 {\ theta}} \ equiv \ tan ^ 2 {\ theta} + 1 [/ matemáticas] y [matemáticas] \ frac {1} {\ sin ^ 2 {\ theta}} \ equiv \ frac {1} {\ tan ^ 2 {\ theta}} + 1 [/ math]
También [math] \ tan {\ theta} \ equiv \ frac {\ sin {\ theta}} {\ cos {\ theta}} [/ math] así que [math] \ sin {\ theta} \ equiv \ cos {\ theta} \ tan {\ theta} [/ math].
[matemáticas] \ begin {align} \ por lo tanto U & = 49 \ int \ frac {\ sin {v} \, \ sqrt {\ tan ^ 2 {v}}} {\ sin ^ 2 {v} \ left (\ frac {49} {\ sin ^ 2 {v}} – \ frac {85} {\ sin ^ 2 {v}} + 85 \ right)} \, \ mathrm dv \\ & = 49 \ int \ frac {\ sin {v} \ tan {v}} {\ sin ^ 2 {v} \ left (\ frac {-36} {\ sin ^ 2 {v}} + 85 \ right)} \, \ mathrm dv \\ & = 49 \ int \ frac {- \ cos {v} \ tan ^ 2 {v}} {\ sin ^ 2 {v} \ left (\ frac {36} {\ sin ^ 2 {v}} – 85 \ right )} \, \ mathrm dv \\ & = 49 \ int \ frac {1} {\ left (\ frac {36} {\ sin ^ 2 {v}} – 85 \ right) \ left (\ frac {1} {\ sin ^ 2 {v}} – 1 \ derecha)} \ izquierda (\ frac {- \ cos {v}} {\ sin ^ 2 {v}} \ derecha) \, \ mathrm dv \ end {align} [/matemáticas]
Establezca [matemáticas] w = \ frac {1} {\ sin {v}} \ \ por lo tanto \ mathrm dw = – \ frac {\ cos {v}} {\ sin ^ 2 {v}} \, \ mathrm dv [ /matemáticas]
[matemáticas] \ begin {align} \ por lo tanto U & = 49 \ int \ frac {1} {\ left (36w ^ 2 – 85 \ right) \ left (w ^ 2 – 1 \ right)} \, \ mathrm dw \\ & = 49 \ int \ frac {1} {\ left (6w + \ sqrt {85} \ right) \ left (6w – \ sqrt {85} \ right) \ left (w + 1 \ right) \ left (w – 1 \ right)} \, \ mathrm dw \\ & = {\ scriptsize \ frac {18} {\ sqrt {85}}} \ int \ frac {-1} {6w + \ sqrt {85}} + \ frac {1} {6w – \ sqrt {85}} \, \ mathrm dw + {\ scriptsize \ frac {1} {2}} \ int \ frac {1} {w + 1} – \ frac {1 } {w – 1} \, \ mathrm dw \\ & = {\ scriptsize \ frac {3} {\ sqrt {85}}} \ left (\ log {\ left \ lvert 6w – \ sqrt {85} \ right \ rvert} – \ log {\ left \ lvert 6w + \ sqrt {85} \ right \ rvert} \ right) + {\ scriptsize \ frac {1} {2}} \ left (\ log {\ left \ lvert w + 1 \ right \ rvert} – \ log {\ left \ lvert w – 1 \ right \ rvert} \ right) \ end {align} [/ math]
Ahora [matemáticas] u = \ frac {7} {2 \ cos {v}} \ \ por lo tanto \ cos {v} = \ frac {7} {2u} [/ matemáticas]
[matemáticas] \ por lo tanto \ sin {v} = \ sqrt {1 – \ left (\ frac {7} {2u} \ right) ^ 2} = \ frac {1} {2u} \ sqrt {4u ^ 2-49 }[/matemáticas]
[matemáticas] \ por lo tanto w = \ frac {1} {\ sin {v}} = \ frac {2u} {\ sqrt {4u ^ 2 – 49}} [/ matemáticas]
Pero [matemáticas] u = x + \ frac {7} {2} [/ matemáticas]
[matemáticas] \ por lo tanto w = \ frac {2x + 7} {\ sqrt {(2x + 7) ^ 2 – 49}} = \ frac {2x + 7} {2 \ sqrt {x (x + 7)}} [/matemáticas]
[matemáticas] \ begin {align} \ por lo tanto U & = {\ scriptsize \ frac {3} {\ sqrt {85}}} \ left (\ log {\ left \ lvert \ frac {6 (2x + 7)} { 2 \ sqrt {x (x + 7)}} – \ sqrt {85} \ right \ rvert} – \ log {\ left \ lvert \ frac {6 (2x + 7)} {2 \ sqrt {x (x + 7)}} + \ sqrt {85} \ right \ rvert} \ right) \\ & \ \ \ + {\ scriptsize \ frac {1} {2}} \ left (\ log {\ left \ lvert \ frac { 2x + 7} {2 \ sqrt {x (x + 7)}} + 1 \ right \ rvert} – \ log {\ left \ lvert \ frac {2x + 7} {2 \ sqrt {x (x + 7) }} – 1 \ right \ rvert} \ right) \\ & = {\ scriptsize \ frac {3} {\ sqrt {85}}} \ log {\ frac {\ left \ lvert \ frac {6 (2x + 7 )} {2 \ sqrt {x (x + 7)}} – \ sqrt {85} \ right \ rvert} {\ left \ lvert \ frac {6 (2x + 7)} {2 \ sqrt {x (x + 7)}} + \ sqrt {85} \ right \ rvert}} + {\ scriptsize \ frac {1} {2}} \ log {\ frac {\ left \ lvert \ frac {2x + 7} {2 \ sqrt {x (x + 7)}} + 1 \ right \ rvert} {\ left \ lvert \ frac {2x + 7} {2 \ sqrt {x (x + 7)}} – 1 \ right \ rvert}} \ \ & = {\ scriptsize \ frac {3} {\ sqrt {85}}} \ log {\ frac {\ left \ lvert 6x + 21 – \ sqrt {85x (x + 7)} \ right \ rvert} {\ left \ lvert 6x + 21 + \ sqrt {85x (x + 7)} \ right \ rvert}} + {\ scriptsize \ frac {1 } {2}} \ log {\ frac {\ left \ lvert 2x + 7 + \ sqrt {4x (x + 7)} \ right \ rvert} {\ left \ lvert 2x + 7 – \ sqrt {4x (x + 7)} \ right \ rvert}} \ end {align} [/ math]
[matemáticas] \ begin {align} V & = \ int \ frac {1} {x ^ 2 + 7x – 9} \, \ mathrm dx \\ & = \ int \ frac {1} {\ left (x + \ frac {1} {2} \ left (7 – \ sqrt {85} \ right) \ right) \ left (x + \ frac {1} {2} \ left (7 + \ sqrt {85} \ right) \ derecha)} \, \ mathrm dx \\ & = {\ scriptsize \ frac {1} {\ sqrt {85}}} \ int \ frac {1} {x + \ frac {1} {2} \ left (7 – \ sqrt {85} \ right)} – \ frac {1} {x + \ frac {1} {2} \ left (7 + \ sqrt {85} \ right)} \, \ mathrm dx \\ & = {\ scriptsize \ frac {1} {\ sqrt {85}}} \ left (\ log {\ left \ lvert x + \ frac {1} {2} \ left (7 – \ sqrt {85} \ right) \ right \ rvert} – \ log {\ left \ lvert x + \ frac {1} {2} \ left (7 + \ sqrt {85} \ right) \ right \ rvert} \ right) \\ & = {\ scriptsize \ frac {1} {\ sqrt {85}}} \ log {\ frac {\ left \ lvert 2x + 7 – \ sqrt {85} \ right \ rvert} {\ left \ lvert 2x + 7 + \ sqrt {85 } \ right \ rvert}} \ end {align} [/ math]
[matemáticas] \ begin {align} T & = U – 3V \\ & = {\ scriptsize \ frac {3} {\ sqrt {85}}} \ log {\ frac {\ left \ lvert 6x + 21 – \ sqrt {85x (x + 7)} \ right \ rvert} {\ left \ lvert 6x + 21 + \ sqrt {85x (x + 7)} \ right \ rvert}} + {\ scriptsize \ frac {1} {2} } \ log {\ frac {\ left \ lvert 2x + 7 + \ sqrt {4x (x + 7)} \ right \ rvert} {\ left \ lvert 2x + 7 – \ sqrt {4x (x + 7)} \ right \ rvert}} \\ & \ \ \ – {\ scriptsize \ frac {3} {\ sqrt {85}}} \ log {\ frac {\ left \ lvert 2x + 7 – \ sqrt {85} \ right \ rvert} {\ left \ lvert 2x + 7 + \ sqrt {85} \ right \ rvert}} \\ & = \ boxed {{\ scriptsize \ frac {3} {\ sqrt {85}}} \ log {\ frac {\ left \ lvert \ left (6x + 21 – \ sqrt {85x (x + 7)} \ right) \ left (2x + 7 + \ sqrt {85} \ right) \ right \ rvert} {\ left \ lvert \ left (6x + 21 + \ sqrt {85x (x + 7)} \ right) \ left (2x + 7 – \ sqrt {85} \ right) \ right \ rvert}} \\ \ \ + {\ scriptsize \ frac {1} {2}} \ log {\ frac {\ left \ lvert 2x + 7 + \ sqrt {4x (x + 7)} \ right \ rvert} {\ left \ lvert 2x + 7 – \ sqrt {4x (x + 7)} \ right \ rvert}} + C} \ end {align} [/ math]