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Resolución de integrales/ cosbx/ cosax/ (cosaxcosbx)^2

Integral de (cosaxcosbx)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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 |  (cos(a*x)*cos(b*x))  dx
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0
∫01(cos⁡(ax)cos⁡(bx))2 dx\int\limits_{0}^{1} \left(\cos{\left(a x \right)} \cos{\left(b x \right)}\right)^{2}\, dx 
Integral((cos(a*x)*cos(b*x))^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
                                 //                                                                                                                                                                                                                         x                                                                                                                                                                                                                           for And(a = 0, b = 0)\
                                 ||                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
                                 ||                                                                                                                                                                                                        2             2                                                                                                                                                                                                                                                   |
                                 ||                                                                                                                                                                                                   x*cos (b*x)   x*sin (b*x)   cos(b*x)*sin(b*x)                                                                                                                                                                                                                          |
                                 ||                                                                                                                                                                                                   ----------- + ----------- + -----------------                                                                                                                                                                                                           for a = 0      |
                                 ||                                                                                                                                                                                                        2             2               2*b                                                                                                                                                                                                                                 |
                                 ||                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
                                 ||                                                                                                                                                                              4               4               2         2             3                      3                                                                                                                                                                                                            |
  /                              ||                                                                                                                                                                       3*x*cos (b*x)   3*x*sin (b*x)   3*x*cos (b*x)*sin (b*x)   3*sin (b*x)*cos(b*x)   5*cos (b*x)*sin(b*x)                                                                                                                                                                                              |
 |                               ||                                                                                                                                                                       ------------- + ------------- + ----------------------- + -------------------- + --------------------                                                                                                                                                                         for Or(a = -b, a = b)|
 |                    2          ||                                                                                                                                                                             8               8                    4                      8*b                    8*b                                                                                                                                                                                                       |
 | (cos(a*x)*cos(b*x))  dx = C + |<                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
 |                               ||                                                                                                                                                                                                        2             2                                                                                                                                                                                                                                                   |
/                                ||                                                                                                                                                                                                   x*cos (a*x)   x*sin (a*x)   cos(a*x)*sin(a*x)                                                                                                                                                                                                                          |
                                 ||                                                                                                                                                                                                   ----------- + ----------- + -----------------                                                                                                                                                                                                           for b = 0      |
                                 ||                                                                                                                                                                                                        2             2               2*a                                                                                                                                                                                                                                 |
                                 ||                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
                                 || 3    2                           3    2                           3    2                           3    2                               3    2         2             3    2         2             3    2         2             3    2         2             3    2         2             3    2         2             3    2         2             3    2         2             2    2                               2    2                                              |
                                 ||a *cos (a*x)*cos(b*x)*sin(b*x)   a *sin (a*x)*cos(b*x)*sin(b*x)   b *cos (b*x)*cos(a*x)*sin(a*x)   b *sin (b*x)*cos(a*x)*sin(a*x)   b*x*a *cos (a*x)*cos (b*x)   b*x*a *cos (a*x)*sin (b*x)   b*x*a *cos (b*x)*sin (a*x)   b*x*a *sin (a*x)*sin (b*x)   a*x*b *cos (a*x)*cos (b*x)   a*x*b *cos (a*x)*sin (b*x)   a*x*b *cos (b*x)*sin (a*x)   a*x*b *sin (a*x)*sin (b*x)   2*a*b *cos (a*x)*cos(b*x)*sin(b*x)   2*b*a *cos (b*x)*cos(a*x)*sin(a*x)                       |
                                 ||------------------------------ + ------------------------------ - ------------------------------ - ------------------------------ + -------------------------- + -------------------------- + -------------------------- + -------------------------- - -------------------------- - -------------------------- - -------------------------- - -------------------------- - ---------------------------------- + ----------------------------------        otherwise      |
                                 ||             3        3                       3        3                       3        3                       3        3                     3        3                   3        3                   3        3                   3        3                   3        3                   3        3                   3        3                   3        3                       3        3                           3        3                                |
                                 ||      - 4*a*b  + 4*b*a                 - 4*a*b  + 4*b*a                 - 4*a*b  + 4*b*a                 - 4*a*b  + 4*b*a               - 4*a*b  + 4*b*a             - 4*a*b  + 4*b*a             - 4*a*b  + 4*b*a             - 4*a*b  + 4*b*a             - 4*a*b  + 4*b*a             - 4*a*b  + 4*b*a             - 4*a*b  + 4*b*a             - 4*a*b  + 4*b*a                 - 4*a*b  + 4*b*a                     - 4*a*b  + 4*b*a                                 |
                                 \\                                                                                                                                                                                                                                                                                                                                                                                                                                                                          /
∫(cos⁡(ax)cos⁡(bx))2 dx=C+{xfor a=0∧b=0xsin⁡2(bx)2+xcos⁡2(bx)2+sin⁡(bx)cos⁡(bx)2bfor a=03xsin⁡4(bx)8+3xsin⁡2(bx)cos⁡2(bx)4+3xcos⁡4(bx)8+3sin⁡3(bx)cos⁡(bx)8b+5sin⁡(bx)cos⁡3(bx)8bfor a=−b∨a=bxsin⁡2(ax)2+xcos⁡2(ax)2+sin⁡(ax)cos⁡(ax)2afor b=0a3bxsin⁡2(ax)sin⁡2(bx)4a3b−4ab3+a3bxsin⁡2(ax)cos⁡2(bx)4a3b−4ab3+a3bxsin⁡2(bx)cos⁡2(ax)4a3b−4ab3+a3bxcos⁡2(ax)cos⁡2(bx)4a3b−4ab3+a3sin⁡2(ax)sin⁡(bx)cos⁡(bx)4a3b−4ab3+a3sin⁡(bx)cos⁡2(ax)cos⁡(bx)4a3b−4ab3+2a2bsin⁡(ax)cos⁡(ax)cos⁡2(bx)4a3b−4ab3−ab3xsin⁡2(ax)sin⁡2(bx)4a3b−4ab3−ab3xsin⁡2(ax)cos⁡2(bx)4a3b−4ab3−ab3xsin⁡2(bx)cos⁡2(ax)4a3b−4ab3−ab3xcos⁡2(ax)cos⁡2(bx)4a3b−4ab3−2ab2sin⁡(bx)cos⁡2(ax)cos⁡(bx)4a3b−4ab3−b3sin⁡(ax)sin⁡2(bx)cos⁡(ax)4a3b−4ab3−b3sin⁡(ax)cos⁡(ax)cos⁡2(bx)4a3b−4ab3otherwise\int \left(\cos{\left(a x \right)} \cos{\left(b x \right)}\right)^{2}\, dx = C + \begin{cases} x & \text{for}\: a = 0 \wedge b = 0 \\\frac{x \sin^{2}{\left(b x \right)}}{2} + \frac{x \cos^{2}{\left(b x \right)}}{2} + \frac{\sin{\left(b x \right)} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = 0 \\\frac{3 x \sin^{4}{\left(b x \right)}}{8} + \frac{3 x \sin^{2}{\left(b x \right)} \cos^{2}{\left(b x \right)}}{4} + \frac{3 x \cos^{4}{\left(b x \right)}}{8} + \frac{3 \sin^{3}{\left(b x \right)} \cos{\left(b x \right)}}{8 b} + \frac{5 \sin{\left(b x \right)} \cos^{3}{\left(b x \right)}}{8 b} & \text{for}\: a = - b \vee a = b \\\frac{x \sin^{2}{\left(a x \right)}}{2} + \frac{x \cos^{2}{\left(a x \right)}}{2} + \frac{\sin{\left(a x \right)} \cos{\left(a x \right)}}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b x \sin^{2}{\left(a x \right)} \sin^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b x \sin^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b x \sin^{2}{\left(b x \right)} \cos^{2}{\left(a x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b x \cos^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin^{2}{\left(a x \right)} \sin{\left(b x \right)} \cos{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin{\left(b x \right)} \cos^{2}{\left(a x \right)} \cos{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{2 a^{2} b \sin{\left(a x \right)} \cos{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \sin^{2}{\left(a x \right)} \sin^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \sin^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \sin^{2}{\left(b x \right)} \cos^{2}{\left(a x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \cos^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{2 a b^{2} \sin{\left(b x \right)} \cos^{2}{\left(a x \right)} \cos{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a x \right)} \sin^{2}{\left(b x \right)} \cos{\left(a x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a x \right)} \cos{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} & \text{otherwise} \end{cases}
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    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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