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Resolución de integrales/ (1+x)/ cos(6x)/ (1+x)^(1/2)cos(6x)

Integral de (1+x)^(1/2)cos(6x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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∫01x+1cos⁡(6x) dx\int\limits_{0}^{1} \sqrt{x + 1} \cos{\left(6 x \right)}\, dx 
Integral(sqrt(1 + x)*cos(6*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                              /                                                _                                                                             _                                      \                                                                                   
                                              |  ___        3/2                               |_  /   1/4, 3/4   |           2\       ___        5/2                        |_  /   3/4, 5/4   |           2\       |                                                                                   
                                              |\/ 3 *(1 + x)   *cos(6)*Gamma(1/4)*Gamma(3/4)* |   |              | -9*(1 + x) |   3*\/ 3 *(1 + x)   *Gamma(3/4)*Gamma(5/4)* |   |              | -9*(1 + x) |*sin(6)|                        /        /    ___   _______\    /    ___   _______\       \
                                   ___   ____ |                                              2  3 \1/2, 5/4, 7/4 |            /                                            2  3 \3/2, 7/4, 9/4 |            /       |     ___   ____         |        |2*\/ 3 *\/ 1 + x |    |2*\/ 3 *\/ 1 + x |       |
  /                            2*\/ 3 *\/ pi *|-------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------|   \/ 3 *\/ pi *(1 + x)*|cos(6)*C|-----------------| + S|-----------------|*sin(6)|
 |                                            |                             ____                                                                                ____                                                |                        |        |        ____     |    |        ____     |       |
 |   _______                                  \                         8*\/ pi *Gamma(5/4)*Gamma(7/4)                                                      4*\/ pi *Gamma(7/4)*Gamma(9/4)                          /                        \        \      \/ pi      /    \      \/ pi      /       /
 | \/ 1 + x *cos(6*x) dx = C - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------
 |                                                                                                                       3                                                                                                                                     3                                        
/
∫x+1cos⁡(6x) dx=C+3π(x+1)(cos⁡(6)C(23x+1π)+sin⁡(6)S(23x+1π))3−23π(33(x+1)52sin⁡(6)Γ(34)Γ(54)2F3(34,5432,74,94|−9(x+1)2)4πΓ(74)Γ(94)+3(x+1)32cos⁡(6)Γ(14)Γ(34)2F3(14,3412,54,74|−9(x+1)2)8πΓ(54)Γ(74))3\int \sqrt{x + 1} \cos{\left(6 x \right)}\, dx = C + \frac{\sqrt{3} \sqrt{\pi} \left(x + 1\right) \left(\cos{\left(6 \right)} C\left(\frac{2 \sqrt{3} \sqrt{x + 1}}{\sqrt{\pi}}\right) + \sin{\left(6 \right)} S\left(\frac{2 \sqrt{3} \sqrt{x + 1}}{\sqrt{\pi}}\right)\right)}{3} - \frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{3 \sqrt{3} \left(x + 1\right)^{\frac{5}{2}} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- 9 \left(x + 1\right)^{2}} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} + \frac{\sqrt{3} \left(x + 1\right)^{\frac{3}{2}} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- 9 \left(x + 1\right)^{2}} \right)}}{8 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}\right)}{3}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.902.5-2.5
Trazar el gráfico f(x)
Respuesta [src] 
                 /                                _                                                                      _                       \                                                                                                                                  /                                     _                                                         _                             \
                 |    ___                        |_  /   3/4, 5/4   |    \            ___                               |_  /   1/4, 3/4   |    \|                                                                                                                                  |  ___                               |_  /   1/4, 3/4   |   \       ___                        |_  /   3/4, 5/4   |   \       |
                 |3*\/ 6 *Gamma(3/4)*Gamma(5/4)* |   |              | -36|*sin(6)   \/ 6 *cos(6)*Gamma(1/4)*Gamma(3/4)* |   |              | -36||                /        /    ___\    /    ___\       \                  /        /    ___\    /    ___\       \                  |\/ 3 *cos(6)*Gamma(1/4)*Gamma(3/4)* |   |              | -9|   3*\/ 3 *Gamma(3/4)*Gamma(5/4)* |   |              | -9|*sin(6)|
      ___   ____ |                              2  3 \3/2, 7/4, 9/4 |    /                                             2  3 \1/2, 5/4, 7/4 |    /|     ___   ____ |        |2*\/ 3 |    |2*\/ 3 |       |       ___   ____ |        |2*\/ 6 |    |2*\/ 6 |       |       ___   ____ |                                   2  3 \1/2, 5/4, 7/4 |   /                                 2  3 \3/2, 7/4, 9/4 |   /       |
  2*\/ 3 *\/ pi *|--------------------------------------------------------------- + -------------------------------------------------------------|   \/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)|   2*\/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)|   2*\/ 3 *\/ pi *|------------------------------------------------------------ + --------------------------------------------------------------|
                 |                    ____                                                              ____                                     |                |        |   ____|    |   ____|       |                  |        |   ____|    |   ____|       |                  |                   ____                                                            ____                                      |
                 \                  \/ pi *Gamma(7/4)*Gamma(9/4)                                    4*\/ pi *Gamma(5/4)*Gamma(7/4)               /                \        \ \/ pi /    \ \/ pi /       /                  \        \ \/ pi /    \ \/ pi /       /                  \               8*\/ pi *Gamma(5/4)*Gamma(7/4)                                  4*\/ pi *Gamma(7/4)*Gamma(9/4)                /
- ------------------------------------------------------------------------------------------------------------------------------------------------ - ---------------------------------------------------- + ------------------------------------------------------ + ----------------------------------------------------------------------------------------------------------------------------------------------
                                                                         3                                                                                                    3                                                       3                                                                                                    3
23π(3cos⁡(6)Γ(14)Γ(34)2F3(14,3412,54,74|−9)8πΓ(54)Γ(74)+33sin⁡(6)Γ(34)Γ(54)2F3(34,5432,74,94|−9)4πΓ(74)Γ(94))3−23π(6cos⁡(6)Γ(14)Γ(34)2F3(14,3412,54,74|−36)4πΓ(54)Γ(74)+36sin⁡(6)Γ(34)Γ(54)2F3(34,5432,74,94|−36)πΓ(74)Γ(94))3−3π(sin⁡(6)S(23π)+cos⁡(6)C(23π))3+23π(sin⁡(6)S(26π)+cos⁡(6)C(26π))3\frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{3} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-9} \right)}}{8 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{3} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-9} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{6} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-36} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{6} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-36} \right)}}{\sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{\sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right)\right)}{3} + \frac{2 \sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right)\right)}{3} 
=
= 
                 /                                _                                                                      _                       \                                                                                                                                  /                                     _                                                         _                             \
                 |    ___                        |_  /   3/4, 5/4   |    \            ___                               |_  /   1/4, 3/4   |    \|                                                                                                                                  |  ___                               |_  /   1/4, 3/4   |   \       ___                        |_  /   3/4, 5/4   |   \       |
                 |3*\/ 6 *Gamma(3/4)*Gamma(5/4)* |   |              | -36|*sin(6)   \/ 6 *cos(6)*Gamma(1/4)*Gamma(3/4)* |   |              | -36||                /        /    ___\    /    ___\       \                  /        /    ___\    /    ___\       \                  |\/ 3 *cos(6)*Gamma(1/4)*Gamma(3/4)* |   |              | -9|   3*\/ 3 *Gamma(3/4)*Gamma(5/4)* |   |              | -9|*sin(6)|
      ___   ____ |                              2  3 \3/2, 7/4, 9/4 |    /                                             2  3 \1/2, 5/4, 7/4 |    /|     ___   ____ |        |2*\/ 3 |    |2*\/ 3 |       |       ___   ____ |        |2*\/ 6 |    |2*\/ 6 |       |       ___   ____ |                                   2  3 \1/2, 5/4, 7/4 |   /                                 2  3 \3/2, 7/4, 9/4 |   /       |
  2*\/ 3 *\/ pi *|--------------------------------------------------------------- + -------------------------------------------------------------|   \/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)|   2*\/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)|   2*\/ 3 *\/ pi *|------------------------------------------------------------ + --------------------------------------------------------------|
                 |                    ____                                                              ____                                     |                |        |   ____|    |   ____|       |                  |        |   ____|    |   ____|       |                  |                   ____                                                            ____                                      |
                 \                  \/ pi *Gamma(7/4)*Gamma(9/4)                                    4*\/ pi *Gamma(5/4)*Gamma(7/4)               /                \        \ \/ pi /    \ \/ pi /       /                  \        \ \/ pi /    \ \/ pi /       /                  \               8*\/ pi *Gamma(5/4)*Gamma(7/4)                                  4*\/ pi *Gamma(7/4)*Gamma(9/4)                /
- ------------------------------------------------------------------------------------------------------------------------------------------------ - ---------------------------------------------------- + ------------------------------------------------------ + ----------------------------------------------------------------------------------------------------------------------------------------------
                                                                         3                                                                                                    3                                                       3                                                                                                    3
23π(3cos⁡(6)Γ(14)Γ(34)2F3(14,3412,54,74|−9)8πΓ(54)Γ(74)+33sin⁡(6)Γ(34)Γ(54)2F3(34,5432,74,94|−9)4πΓ(74)Γ(94))3−23π(6cos⁡(6)Γ(14)Γ(34)2F3(14,3412,54,74|−36)4πΓ(54)Γ(74)+36sin⁡(6)Γ(34)Γ(54)2F3(34,5432,74,94|−36)πΓ(74)Γ(94))3−3π(sin⁡(6)S(23π)+cos⁡(6)C(23π))3+23π(sin⁡(6)S(26π)+cos⁡(6)C(26π))3\frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{3} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-9} \right)}}{8 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{3} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-9} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{6} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-36} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{6} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-36} \right)}}{\sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{\sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right)\right)}{3} + \frac{2 \sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right)\right)}{3} 
-2*sqrt(3)*sqrt(pi)*(3*sqrt(6)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -36)*sin(6)/(sqrt(pi)*gamma(7/4)*gamma(9/4)) + sqrt(6)*cos(6)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -36)/(4*sqrt(pi)*gamma(5/4)*gamma(7/4)))/3 - sqrt(3)*sqrt(pi)*(cos(6)*fresnelc(2*sqrt(3)/sqrt(pi)) + fresnels(2*sqrt(3)/sqrt(pi))*sin(6))/3 + 2*sqrt(3)*sqrt(pi)*(cos(6)*fresnelc(2*sqrt(6)/sqrt(pi)) + fresnels(2*sqrt(6)/sqrt(pi))*sin(6))/3 + 2*sqrt(3)*sqrt(pi)*(sqrt(3)*cos(6)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -9)/(8*sqrt(pi)*gamma(5/4)*gamma(7/4)) + 3*sqrt(3)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -9)*sin(6)/(4*sqrt(pi)*gamma(7/4)*gamma(9/4)))/3
Respuesta numérica [src] 
-0.0702330998031601
-0.0702330998031601

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

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