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Resolución de integrales/ arctan/ t^2(arctant)/(1+t^2)

Integral de t^2(arctant)/(1+t^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  3              
  /              
 |               
 |   2           
 |  t *atan(t)   
 |  ---------- dt
 |         2     
 |    1 + t      
 |               
/                
0
∫03t2atan⁡(t)t2+1 dt\int\limits_{0}^{3} \frac{t^{2} \operatorname{atan}{\left(t \right)}}{t^{2} + 1}\, dt 
Integral((t^2*atan(t))/(1 + t^2), (t, 0, 3))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=tan(_theta)**2*atan(tan(_theta)), substep=PartsRule(u=atan(tan(_theta)), dv=tan(_theta)**2, v_step=RewriteRule(rewritten=sec(_theta)**2 - 1, substep=AddRule(substeps=[TrigRule(func='sec**2', arg=_theta, context=sec(_theta)**2, symbol=_theta), ConstantRule(constant=-1, context=-1, symbol=_theta)], context=sec(_theta)**2 - 1, symbol=_theta), context=tan(_theta)**2, symbol=_theta), second_step=AddRule(substeps=[ConstantTimesRule(constant=-1, other=_theta, substep=PowerRule(base=_theta, exp=1, context=_theta, symbol=_theta), context=-_theta, symbol=_theta), RewriteRule(rewritten=sin(_theta)/cos(_theta), substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=1/_u, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=1/_u, symbol=_u), context=sin(_theta)/cos(_theta), symbol=_theta), context=tan(_theta), symbol=_theta)], context=-_theta + tan(_theta), symbol=_theta), context=tan(_theta)**2*atan(tan(_theta)), symbol=_theta), restriction=True, context=(t**2*atan(t))/(t**2 + 1), symbol=t)

  1. Ahora simplificar:

    tatan⁡(t)−log⁡(t2+1)2−atan⁡2(t)2t \operatorname{atan}{\left(t \right)} - \frac{\log{\left(t^{2} + 1 \right)}}{2} - \frac{\operatorname{atan}^{2}{\left(t \right)}}{2}

  2. Añadimos la constante de integración:

    tatan⁡(t)−log⁡(t2+1)2−atan⁡2(t)2+constantt \operatorname{atan}{\left(t \right)} - \frac{\log{\left(t^{2} + 1 \right)}}{2} - \frac{\operatorname{atan}^{2}{\left(t \right)}}{2}+ \mathrm{constant}

Respuesta:

tatan⁡(t)−log⁡(t2+1)2−atan⁡2(t)2+constantt \operatorname{atan}{\left(t \right)} - \frac{\log{\left(t^{2} + 1 \right)}}{2} - \frac{\operatorname{atan}^{2}{\left(t \right)}}{2}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                       
 |                                                                        
 |  2                      2                                              
 | t *atan(t)          atan (t)                              /     1     \
 | ---------- dt = C + -------- + (t - atan(t))*atan(t) + log|-----------|
 |        2               2                                  |   ________|
 |   1 + t                                                   |  /      2 |
 |                                                           \\/  1 + t  /
/
∫t2atan⁡(t)t2+1 dt=C+(t−atan⁡(t))atan⁡(t)+log⁡(1t2+1)+atan⁡2(t)2\int \frac{t^{2} \operatorname{atan}{\left(t \right)}}{t^{2} + 1}\, dt = C + \left(t - \operatorname{atan}{\left(t \right)}\right) \operatorname{atan}{\left(t \right)} + \log{\left(\frac{1}{\sqrt{t^{2} + 1}} \right)} + \frac{\operatorname{atan}^{2}{\left(t \right)}}{2}
Simplificar
Gráfica
0.003.000.250.500.751.001.251.501.752.002.252.502.7502
Trazar el gráfico f(x)
Respuesta [src] 
                2             
            atan (3)   log(10)
3*atan(3) - -------- - -------
               2          2
−log⁡(10)2−atan⁡2(3)2+3atan⁡(3)- \frac{\log{\left(10 \right)}}{2} - \frac{\operatorname{atan}^{2}{\left(3 \right)}}{2} + 3 \operatorname{atan}{\left(3 \right)} 
=
= 
                2             
            atan (3)   log(10)
3*atan(3) - -------- - -------
               2          2
−log⁡(10)2−atan⁡2(3)2+3atan⁡(3)- \frac{\log{\left(10 \right)}}{2} - \frac{\operatorname{atan}^{2}{\left(3 \right)}}{2} + 3 \operatorname{atan}{\left(3 \right)} 
3*atan(3) - atan(3)^2/2 - log(10)/2
Respuesta numérica [src] 
1.81578709992476
1.81578709992476

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

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