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Resolución de integrales/ dx/(a+bsinx)

Integral de dx/(a+bsinx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |  a + b*sin(x)   
 |                 
/                  
0
∫011a+bsin⁡(x) dx\int\limits_{0}^{1} \frac{1}{a + b \sin{\left(x \right)}}\, dx 
Integral(1/(a + b*sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
                         //                                            /   /x\\                                                            \
                         ||                                     zoo*log|tan|-||                                       for And(a = 0, b = 0)|
                         ||                                            \   \2//                                                            |
                         ||                                                                                                                |
                         ||                                            2                                                                   |
                         ||                                      -------------                                             for a = -b      |
                         ||                                                /x\                                                             |
                         ||                                      -b + b*tan|-|                                                             |
                         ||                                                \2/                                                             |
                         ||                                                                                                                |
                         ||                                          -2                                                                    |
                         ||                                      ------------                                               for a = b      |
  /                      ||                                               /x\                                                              |
 |                       ||                                      b + b*tan|-|                                                              |
 |      1                ||                                               \2/                                                              |
 | ------------ dx = C + |<                                                                                                                |
 | a + b*sin(x)          ||                                          /   /x\\                                                              |
 |                       ||                                       log|tan|-||                                                              |
/                        ||                                          \   \2//                                                              |
                         ||                                       -----------                                               for a = 0      |
                         ||                                            b                                                                   |
                         ||                                                                                                                |
                         ||                /       _________         \                   /       _________         \                       |
                         ||   _________    |      /  2    2          |      _________    |      /  2    2          |                       |
                         ||  /  2    2     |b   \/  b  - a        /x\|     /  2    2     |b   \/  b  - a        /x\|                       |
                         ||\/  b  - a  *log|- + ------------ + tan|-||   \/  b  - a  *log|- - ------------ + tan|-||                       |
                         ||                \a        a            \2//                   \a        a            \2//                       |
                         ||------------------------------------------- - -------------------------------------------        otherwise      |
                         ||                   2    2                                        2    2                                         |
                         ||                  a  - b                                        a  - b                                          |
                         \\                                                                                                                /
∫1a+bsin⁡(x) dx=C+{∞~log⁡(tan⁡(x2))for a=0∧b=02btan⁡(x2)−bfor a=−b−2btan⁡(x2)+bfor a=blog⁡(tan⁡(x2))bfor a=0−−a2+b2log⁡(tan⁡(x2)+ba−−a2+b2a)a2−b2+−a2+b2log⁡(tan⁡(x2)+ba+−a2+b2a)a2−b2otherwise\int \frac{1}{a + b \sin{\left(x \right)}}\, dx = C + \begin{cases} \tilde{\infty} \log{\left(\tan{\left(\frac{x}{2} \right)} \right)} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2}{b \tan{\left(\frac{x}{2} \right)} - b} & \text{for}\: a = - b \\- \frac{2}{b \tan{\left(\frac{x}{2} \right)} + b} & \text{for}\: a = b \\\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{b} & \text{for}\: a = 0 \\- \frac{\sqrt{- a^{2} + b^{2}} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{a^{2} - b^{2}} + \frac{\sqrt{- a^{2} + b^{2}} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                                                            ____                                                                                                                                                                          
|                                                           /  2                                                            /   /               ____\     /        ____         ____\     /               ____         ____\          ____\
|                                            2          2*\/  b                                                             |   |              /  2 |     |       /  2         /  2 |     |              /  2         /  2 |         /  2 |
|                                            - + -----------------------                                              for Or\And\a = 0, a = -\/  b  /, And\a = -\/  b  , a = \/  b  /, And\a = 0, a = -\/  b  , a = \/  b  /, a = -\/  b  /
|                                            b                      ____                                                                                                                                                                   
|                                                 2                /  2                                                                                                                                                                    
|                                                b *tan(1/2) - b*\/  b                                                                                                                                                                     
|                                                                                                                                                                                                                                          
|                                                            ____                                                                                                                                                                          
|                                                           /  2                                                                                                /   /              ____\         ____\                                     
|                                            2          2*\/  b                                                                                                 |   |             /  2 |        /  2 |                                     
|                                            - - -----------------------                                                                                  for Or\And\a = 0, a = \/  b  /, a = \/  b  /                                     
|                                            b        ____                                                                                                                                                                                 
|                                                    /  2     2                                                                                                                                                                            
<                                                b*\/  b   + b *tan(1/2)                                                                                                                                                                   
|                                                                                                                                                                                                                                          
|                                                   /1\   log(tan(1/2))                                                                                                                                                                    
|                                            oo*sign|-| + -------------                                                                                                     for a = 0                                                      
|                                                   \b/         b                                                                                                                                                                          
|                                                                                                                                                                                                                                          
|   /       _________\      /       _________           \      /       _________\      /       _________           \                                                                                                                       
|   |      /  2    2 |      |      /  2    2            |      |      /  2    2 |      |      /  2    2            |                                                                                                                       
|   |b   \/  b  - a  |      |b   \/  b  - a             |      |b   \/  b  - a  |      |b   \/  b  - a             |                                                                                                                       
|log|- + ------------|   log|- - ------------ + tan(1/2)|   log|- - ------------|   log|- + ------------ + tan(1/2)|                                                                                                                       
|   \a        a      /      \a        a                 /      \a        a      /      \a        a                 /                                                                                                                       
|--------------------- + -------------------------------- - --------------------- - --------------------------------                                                        otherwise                                                      
|        _________                    _________                     _________                    _________                                                                                                                                 
|       /  2    2                    /  2    2                     /  2    2                    /  2    2                                                                                                                                  
\     \/  b  - a                   \/  b  - a                    \/  b  - a                   \/  b  - a
{2b2b2tan⁡(12)−bb2+2bfor (a=0∧a=−b2)∨(a=−b2∧a=b2)∨(a=0∧a=−b2∧a=b2)∨a=−b2−2b2b2tan⁡(12)+bb2+2bfor (a=0∧a=b2)∨a=b2∞sign⁡(1b)+log⁡(tan⁡(12))bfor a=0−log⁡(ba−−a2+b2a)−a2+b2+log⁡(ba+−a2+b2a)−a2+b2+log⁡(tan⁡(12)+ba−−a2+b2a)−a2+b2−log⁡(tan⁡(12)+ba+−a2+b2a)−a2+b2otherwise\begin{cases} \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} - b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - \sqrt{b^{2}}\right) \vee \left(a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee \left(a = 0 \wedge a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee a = - \sqrt{b^{2}} \\- \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} + b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = \sqrt{b^{2}}\right) \vee a = \sqrt{b^{2}} \\\infty \operatorname{sign}{\left(\frac{1}{b} \right)} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} \right)}}{b} & \text{for}\: a = 0 \\- \frac{\log{\left(\frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} & \text{otherwise} \end{cases} 
=
= 
/                                                            ____                                                                                                                                                                          
|                                                           /  2                                                            /   /               ____\     /        ____         ____\     /               ____         ____\          ____\
|                                            2          2*\/  b                                                             |   |              /  2 |     |       /  2         /  2 |     |              /  2         /  2 |         /  2 |
|                                            - + -----------------------                                              for Or\And\a = 0, a = -\/  b  /, And\a = -\/  b  , a = \/  b  /, And\a = 0, a = -\/  b  , a = \/  b  /, a = -\/  b  /
|                                            b                      ____                                                                                                                                                                   
|                                                 2                /  2                                                                                                                                                                    
|                                                b *tan(1/2) - b*\/  b                                                                                                                                                                     
|                                                                                                                                                                                                                                          
|                                                            ____                                                                                                                                                                          
|                                                           /  2                                                                                                /   /              ____\         ____\                                     
|                                            2          2*\/  b                                                                                                 |   |             /  2 |        /  2 |                                     
|                                            - - -----------------------                                                                                  for Or\And\a = 0, a = \/  b  /, a = \/  b  /                                     
|                                            b        ____                                                                                                                                                                                 
|                                                    /  2     2                                                                                                                                                                            
<                                                b*\/  b   + b *tan(1/2)                                                                                                                                                                   
|                                                                                                                                                                                                                                          
|                                                   /1\   log(tan(1/2))                                                                                                                                                                    
|                                            oo*sign|-| + -------------                                                                                                     for a = 0                                                      
|                                                   \b/         b                                                                                                                                                                          
|                                                                                                                                                                                                                                          
|   /       _________\      /       _________           \      /       _________\      /       _________           \                                                                                                                       
|   |      /  2    2 |      |      /  2    2            |      |      /  2    2 |      |      /  2    2            |                                                                                                                       
|   |b   \/  b  - a  |      |b   \/  b  - a             |      |b   \/  b  - a  |      |b   \/  b  - a             |                                                                                                                       
|log|- + ------------|   log|- - ------------ + tan(1/2)|   log|- - ------------|   log|- + ------------ + tan(1/2)|                                                                                                                       
|   \a        a      /      \a        a                 /      \a        a      /      \a        a                 /                                                                                                                       
|--------------------- + -------------------------------- - --------------------- - --------------------------------                                                        otherwise                                                      
|        _________                    _________                     _________                    _________                                                                                                                                 
|       /  2    2                    /  2    2                     /  2    2                    /  2    2                                                                                                                                  
\     \/  b  - a                   \/  b  - a                    \/  b  - a                   \/  b  - a
{2b2b2tan⁡(12)−bb2+2bfor (a=0∧a=−b2)∨(a=−b2∧a=b2)∨(a=0∧a=−b2∧a=b2)∨a=−b2−2b2b2tan⁡(12)+bb2+2bfor (a=0∧a=b2)∨a=b2∞sign⁡(1b)+log⁡(tan⁡(12))bfor a=0−log⁡(ba−−a2+b2a)−a2+b2+log⁡(ba+−a2+b2a)−a2+b2+log⁡(tan⁡(12)+ba−−a2+b2a)−a2+b2−log⁡(tan⁡(12)+ba+−a2+b2a)−a2+b2otherwise\begin{cases} \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} - b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - \sqrt{b^{2}}\right) \vee \left(a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee \left(a = 0 \wedge a = - \sqrt{b^{2}} \wedge a = \sqrt{b^{2}}\right) \vee a = - \sqrt{b^{2}} \\- \frac{2 \sqrt{b^{2}}}{b^{2} \tan{\left(\frac{1}{2} \right)} + b \sqrt{b^{2}}} + \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = \sqrt{b^{2}}\right) \vee a = \sqrt{b^{2}} \\\infty \operatorname{sign}{\left(\frac{1}{b} \right)} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} \right)}}{b} & \text{for}\: a = 0 \\- \frac{\log{\left(\frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right)}}{\sqrt{- a^{2} + b^{2}}} & \text{otherwise} \end{cases} 
Piecewise((2/b + 2*sqrt(b^2)/(b^2*tan(1/2) - b*sqrt(b^2)), (a = -sqrt(b^2))∨((a = 0)∧(a = -sqrt(b^2)))∨((a = sqrt(b^2))∧(a = -sqrt(b^2)))∨((a = 0)∧(a = sqrt(b^2))∧(a = -sqrt(b^2)))), (2/b - 2*sqrt(b^2)/(b*sqrt(b^2) + b^2*tan(1/2)), (a = sqrt(b^2))∨((a = 0)∧(a = sqrt(b^2)))), (oo*sign(1/b) + log(tan(1/2))/b, a = 0), (log(b/a + sqrt(b^2 - a^2)/a)/sqrt(b^2 - a^2) + log(b/a - sqrt(b^2 - a^2)/a + tan(1/2))/sqrt(b^2 - a^2) - log(b/a - sqrt(b^2 - a^2)/a)/sqrt(b^2 - a^2) - log(b/a + sqrt(b^2 - a^2)/a + tan(1/2))/sqrt(b^2 - a^2), True))

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

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