Sr Examen

Otras calculadoras

Resolución de integrales/ (sin(x)+1)/(sin(x)+2*cos(x)+3)

Integral de (sin(x)+1)/(sin(x)+2*cos(x)+3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                         
  /                         
 |                          
 |        sin(x) + 1        
 |  --------------------- dx
 |  sin(x) + 2*cos(x) + 3   
 |                          
/                           
0
∫01sin⁡(x)+1(sin⁡(x)+2cos⁡(x))+3 dx\int\limits_{0}^{1} \frac{\sin{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 3}\, dx 
Integral((sin(x) + 1)/(sin(x) + 2*cos(x) + 3), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    sin⁡(x)+1(sin⁡(x)+2cos⁡(x))+3=sin⁡(x)(sin⁡(x)+2cos⁡(x))+3+1(sin⁡(x)+2cos⁡(x))+3\frac{\sin{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 3} = \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 3} + \frac{1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 3}

  2. Integramos término a término:

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

      Pero la integral

      x5+2log⁡(tan⁡2(x2)+1)5−2log⁡(tan⁡2(x2)+2tan⁡(x2)+5)5−3atan⁡(tan⁡(x2)2+12)5−3π⌊x2−π2π⌋5\frac{x}{5} + \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{5} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 5 \right)}}{5} - \frac{3 \operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} - \frac{3 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{5}

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

      Pero la integral

      atan⁡(tan⁡(x2)2+12)+π⌊x2−π2π⌋\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor

    El resultado es: x5+2log⁡(tan⁡2(x2)+1)5−2log⁡(tan⁡2(x2)+2tan⁡(x2)+5)5+2atan⁡(tan⁡(x2)2+12)5+2π⌊x2−π2π⌋5\frac{x}{5} + \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{5} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 5 \right)}}{5} + \frac{2 \operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} + \frac{2 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{5}

  3. Ahora simplificar:

    x5+2log⁡(1cos⁡(x)+1)5−2log⁡(tan⁡2(x2)+2tan⁡(x2)+5)5+2atan⁡(tan⁡(x2)2+12)5+2π⌊x2π−12⌋5+2log⁡(2)5\frac{x}{5} + \frac{2 \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)}}{5} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 5 \right)}}{5} + \frac{2 \operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} + \frac{2 \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor}{5} + \frac{2 \log{\left(2 \right)}}{5}

  4. Añadimos la constante de integración:

    x5+2log⁡(1cos⁡(x)+1)5−2log⁡(tan⁡2(x2)+2tan⁡(x2)+5)5+2atan⁡(tan⁡(x2)2+12)5+2π⌊x2π−12⌋5+2log⁡(2)5+constant\frac{x}{5} + \frac{2 \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)}}{5} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 5 \right)}}{5} + \frac{2 \operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} + \frac{2 \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor}{5} + \frac{2 \log{\left(2 \right)}}{5}+ \mathrm{constant}

Respuesta:

x5+2log⁡(1cos⁡(x)+1)5−2log⁡(tan⁡2(x2)+2tan⁡(x2)+5)5+2atan⁡(tan⁡(x2)2+12)5+2π⌊x2π−12⌋5+2log⁡(2)5+constant\frac{x}{5} + \frac{2 \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)}}{5} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 5 \right)}}{5} + \frac{2 \operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} + \frac{2 \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor}{5} + \frac{2 \log{\left(2 \right)}}{5}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                                                            /       /x\\                                  /x   pi\
                                                                            |    tan|-||                                  |- - --|
  /                                    /       2/x\        /x\\             |1      \2/|        /       2/x\\             |2   2 |
 |                                2*log|5 + tan |-| + 2*tan|-||       2*atan|- + ------|   2*log|1 + tan |-||   2*pi*floor|------|
 |       sin(x) + 1                    \        \2/        \2//   x         \2     2   /        \        \2//             \  pi  /
 | --------------------- dx = C - ----------------------------- + - + ------------------ + ------------------ + ------------------
 | sin(x) + 2*cos(x) + 3                        5                 5           5                    5                    5         
 |                                                                                                                                
/
∫sin⁡(x)+1(sin⁡(x)+2cos⁡(x))+3 dx=C+x5+2log⁡(tan⁡2(x2)+1)5−2log⁡(tan⁡2(x2)+2tan⁡(x2)+5)5+2atan⁡(tan⁡(x2)2+12)5+2π⌊x2−π2π⌋5\int \frac{\sin{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 3}\, dx = C + \frac{x}{5} + \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{5} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 5 \right)}}{5} + \frac{2 \operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} + \frac{2 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{5}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.902.5-2.5
Trazar el gráfico f(x)
Respuesta [src] 
                                                            /1   tan(1/2)\                                  
                       /       2                  \   2*atan|- + --------|                   /       2     \
1   2*atan(1/2)   2*log\5 + tan (1/2) + 2*tan(1/2)/         \2      2    /   2*log(5)   2*log\1 + tan (1/2)/
- - ----------- - --------------------------------- + -------------------- + -------- + --------------------
5        5                        5                            5                5                5
−2log⁡(tan⁡2(12)+2tan⁡(12)+5)5−2atan⁡(12)5+2log⁡(tan⁡2(12)+1)5+15+2atan⁡(tan⁡(12)2+12)5+2log⁡(5)5- \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan{\left(\frac{1}{2} \right)} + 5 \right)}}{5} - \frac{2 \operatorname{atan}{\left(\frac{1}{2} \right)}}{5} + \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{5} + \frac{1}{5} + \frac{2 \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} + \frac{2 \log{\left(5 \right)}}{5} 
=
= 
                                                            /1   tan(1/2)\                                  
                       /       2                  \   2*atan|- + --------|                   /       2     \
1   2*atan(1/2)   2*log\5 + tan (1/2) + 2*tan(1/2)/         \2      2    /   2*log(5)   2*log\1 + tan (1/2)/
- - ----------- - --------------------------------- + -------------------- + -------- + --------------------
5        5                        5                            5                5                5
−2log⁡(tan⁡2(12)+2tan⁡(12)+5)5−2atan⁡(12)5+2log⁡(tan⁡2(12)+1)5+15+2atan⁡(tan⁡(12)2+12)5+2log⁡(5)5- \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan{\left(\frac{1}{2} \right)} + 5 \right)}}{5} - \frac{2 \operatorname{atan}{\left(\frac{1}{2} \right)}}{5} + \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{5} + \frac{1}{5} + \frac{2 \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{2} + \frac{1}{2} \right)}}{5} + \frac{2 \log{\left(5 \right)}}{5} 
1/5 - 2*atan(1/2)/5 - 2*log(5 + tan(1/2)^2 + 2*tan(1/2))/5 + 2*atan(1/2 + tan(1/2)/2)/5 + 2*log(5)/5 + 2*log(1 + tan(1/2)^2)/5
Respuesta numérica [src] 
0.28408560237831
0.28408560237831

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

    © Sr Examen – Калькулятор