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Resolución de integrales/ atan(x*x)/x^2

Integral de atan(x*x)/x^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 1/2            
  /             
 |              
 |  atan(x*x)   
 |  --------- dx
 |       2      
 |      x       
 |              
/               
0
∫012atan⁡(xx)x2 dx\int\limits_{0}^{\frac{1}{2}} \frac{\operatorname{atan}{\left(x x \right)}}{x^{2}}\, dx 
Integral(atan(x*x)/x^2, (x, 0, 1/2))
Solución detallada

  1. Usamos la integración por partes:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    que u(x)=atan⁡(xx)u{\left(x \right)} = \operatorname{atan}{\left(x x \right)} y que dv⁡(x)=1x2\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}.

    Entonces du⁡(x)=2xx4+1\operatorname{du}{\left(x \right)} = \frac{2 x}{x^{4} + 1}.

    Para buscar v(x)v{\left(x \right)}:

    1. Integral xnx^{n} es xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫1x2 dx=−1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

    Ahora resolvemos podintegral.

  2. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫(−2x4+1) dx=−2∫1x4+1 dx\int \left(- \frac{2}{x^{4} + 1}\right)\, dx = - 2 \int \frac{1}{x^{4} + 1}\, dx

    1. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

      −2log⁡(x2−2x+1)8+2log⁡(x2+2x+1)8+2atan⁡(2x−1)4+2atan⁡(2x+1)4- \frac{\sqrt{2} \log{\left(x^{2} - \sqrt{2} x + 1 \right)}}{8} + \frac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x - 1 \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4}

    Por lo tanto, el resultado es: 2log⁡(x2−2x+1)4−2log⁡(x2+2x+1)4−2atan⁡(2x−1)2−2atan⁡(2x+1)2\frac{\sqrt{2} \log{\left(x^{2} - \sqrt{2} x + 1 \right)}}{4} - \frac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x - 1 \right)}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{2}

  3. Ahora simplificar:

    2x(−log⁡(x2−2x+1)+log⁡(x2+2x+1)+2atan⁡(2x−1)+2atan⁡(2x+1))4−atan⁡(x2)x\frac{\frac{\sqrt{2} x \left(- \log{\left(x^{2} - \sqrt{2} x + 1 \right)} + \log{\left(x^{2} + \sqrt{2} x + 1 \right)} + 2 \operatorname{atan}{\left(\sqrt{2} x - 1 \right)} + 2 \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}\right)}{4} - \operatorname{atan}{\left(x^{2} \right)}}{x}

  4. Añadimos la constante de integración:

    2x(−log⁡(x2−2x+1)+log⁡(x2+2x+1)+2atan⁡(2x−1)+2atan⁡(2x+1))4−atan⁡(x2)x+constant\frac{\frac{\sqrt{2} x \left(- \log{\left(x^{2} - \sqrt{2} x + 1 \right)} + \log{\left(x^{2} + \sqrt{2} x + 1 \right)} + 2 \operatorname{atan}{\left(\sqrt{2} x - 1 \right)} + 2 \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}\right)}{4} - \operatorname{atan}{\left(x^{2} \right)}}{x}+ \mathrm{constant}

Respuesta:

2x(−log⁡(x2−2x+1)+log⁡(x2+2x+1)+2atan⁡(2x−1)+2atan⁡(2x+1))4−atan⁡(x2)x+constant\frac{\frac{\sqrt{2} x \left(- \log{\left(x^{2} - \sqrt{2} x + 1 \right)} + \log{\left(x^{2} + \sqrt{2} x + 1 \right)} + 2 \operatorname{atan}{\left(\sqrt{2} x - 1 \right)} + 2 \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}\right)}{4} - \operatorname{atan}{\left(x^{2} \right)}}{x}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                                                             
 |                      ___     /        ___\     ___     /         ___\                 ___    /     2       ___\     ___    /     2       ___\
 | atan(x*x)          \/ 2 *atan\1 + x*\/ 2 /   \/ 2 *atan\-1 + x*\/ 2 /   atan(x*x)   \/ 2 *log\1 + x  - x*\/ 2 /   \/ 2 *log\1 + x  + x*\/ 2 /
 | --------- dx = C + ----------------------- + ------------------------ - --------- - --------------------------- + ---------------------------
 |      2                        2                         2                   x                    4                             4             
 |     x                                                                                                                                        
 |                                                                                                                                              
/
∫atan⁡(xx)x2 dx=C−2log⁡(x2−2x+1)4+2log⁡(x2+2x+1)4+2atan⁡(2x−1)2+2atan⁡(2x+1)2−atan⁡(xx)x\int \frac{\operatorname{atan}{\left(x x \right)}}{x^{2}}\, dx = C - \frac{\sqrt{2} \log{\left(x^{2} - \sqrt{2} x + 1 \right)}}{4} + \frac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x - 1 \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{2} - \frac{\operatorname{atan}{\left(x x \right)}}{x}
Simplificar
Gráfica
0.000.500.050.100.150.200.250.300.350.400.450.52.0
Trazar el gráfico f(x)
Respuesta [src] 
                         /      ___\     ___                  ___     ___    /        ___\     ___    /        ___\
                 ___     |    \/ 2 |   \/ 2 *atan(1/4)   pi*\/ 2    \/ 2 *log\5 - 2*\/ 2 /   \/ 2 *log\5 + 2*\/ 2 /
-2*atan(1/4) + \/ 2 *atan|1 + -----| + --------------- - -------- - ---------------------- + ----------------------
                         \      2  /          2             4                 4                        4
−2π4−2atan⁡(14)−2log⁡(5−22)4+2atan⁡(14)2+2log⁡(22+5)4+2atan⁡(22+1)- \frac{\sqrt{2} \pi}{4} - 2 \operatorname{atan}{\left(\frac{1}{4} \right)} - \frac{\sqrt{2} \log{\left(5 - 2 \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{1}{4} \right)}}{2} + \frac{\sqrt{2} \log{\left(2 \sqrt{2} + 5 \right)}}{4} + \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} + 1 \right)} 
=
= 
                         /      ___\     ___                  ___     ___    /        ___\     ___    /        ___\
                 ___     |    \/ 2 |   \/ 2 *atan(1/4)   pi*\/ 2    \/ 2 *log\5 - 2*\/ 2 /   \/ 2 *log\5 + 2*\/ 2 /
-2*atan(1/4) + \/ 2 *atan|1 + -----| + --------------- - -------- - ---------------------- + ----------------------
                         \      2  /          2             4                 4                        4
−2π4−2atan⁡(14)−2log⁡(5−22)4+2atan⁡(14)2+2log⁡(22+5)4+2atan⁡(22+1)- \frac{\sqrt{2} \pi}{4} - 2 \operatorname{atan}{\left(\frac{1}{4} \right)} - \frac{\sqrt{2} \log{\left(5 - 2 \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{1}{4} \right)}}{2} + \frac{\sqrt{2} \log{\left(2 \sqrt{2} + 5 \right)}}{4} + \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} + 1 \right)} 
-2*atan(1/4) + sqrt(2)*atan(1 + sqrt(2)/2) + sqrt(2)*atan(1/4)/2 - pi*sqrt(2)/4 - sqrt(2)*log(5 - 2*sqrt(2))/4 + sqrt(2)*log(5 + 2*sqrt(2))/4
Respuesta numérica [src] 
0.497958775901148
0.497958775901148

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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