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Resolución de integrales/ sqrt(4-5*x)*sinx

Integral de sqrt(4-5*x)*sinx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                      
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 2                       
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 |    _________          
 |  \/ 4 - 5*x *sin(x) dx
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0
∫0π24−5xsin⁡(x) dx\int\limits_{0}^{\frac{\pi}{2}} \sqrt{4 - 5 x} \sin{\left(x \right)}\, dx 
Integral(sqrt(4 - 5*x)*sin(x), (x, 0, pi/2))
Gráfica
0.000.100.200.300.400.500.600.700.801-1
Trazar el gráfico f(x)
Respuesta [src] 
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                                                                                  |                                                  /              |            2 \                                                                     /              |            2 \|                                                                                                                                                                                                                                                                   
                                                                                  |                   3/2                         _  |              |  /    5*pi\  |                             5/2                                  _  |              |  /    5*pi\  ||                                                                                                                                                                                                                                                                   
                                                                                  |    ____ /    5*pi\                           |_  |   1/4, 3/4   | -|4 - ----|  |              ____ /    5*pi\                                    |_  |   3/4, 5/4   | -|4 - ----|  ||                   /                                 _                                                                               _                         \                            /          /           __________\    /           __________\         \
                                                                                  |  \/ 10 *|4 - ----|   *Gamma(1/4)*Gamma(3/4)* |   |              |  \     2  /  |*sin(4/5)   \/ 10 *|4 - ----|   *cos(4/5)*Gamma(3/4)*Gamma(5/4)* |   |              |  \     2  /  ||                   |    ____                        |_  /   1/4, 3/4   |      \                ____                                 |_  /   3/4, 5/4   |      \|                            |          |  ____    /     5*pi |    |  ____    /     5*pi |         |
                  /          /    ____\    /    ____\         \                   |         \     2  /                          2  3 |1/2, 5/4, 7/4 | -------------|                   \     2  /                                   2  3 |3/2, 7/4, 9/4 | -------------||                   |  \/ 10 *Gamma(1/4)*Gamma(3/4)* |   |              | -4/25|*sin(4/5)   2*\/ 10 *cos(4/5)*Gamma(3/4)*Gamma(5/4)* |   |              | -4/25||                            |          |\/ 10 *  /  4 - ---- |    |\/ 10 *  /  4 - ---- |         |
      ____   ____ |          |2*\/ 10 |    |2*\/ 10 |         |       ____   ____ |                                                  \              |      100     /                                                                     \              |      100     /|       ____   ____ |                               2  3 \1/2, 5/4, 7/4 |      /                                                    2  3 \3/2, 7/4, 9/4 |      /|     ____   ____ /    5*pi\ |          |       \/        2   |    |       \/        2   |         |
  4*\/ 10 *\/ pi *|cos(4/5)*S|--------| - C|--------|*sin(4/5)|   2*\/ 10 *\/ pi *|- ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------|   2*\/ 10 *\/ pi *|- ------------------------------------------------------------------ + --------------------------------------------------------------------|   \/ 10 *\/ pi *|4 - ----|*|cos(4/5)*S|---------------------| - C|---------------------|*sin(4/5)|
                  |          |    ____|    |    ____|         |                   |                                   ____                                                                                        ____                                                  |                   |                        ____                                                                  ____                                         |                 \     2  / |          |           ____      |    |           ____      |         |
                  \          \5*\/ pi /    \5*\/ pi /         /                   \                              80*\/ pi *Gamma(5/4)*Gamma(7/4)                                                            400*\/ pi *Gamma(7/4)*Gamma(9/4)                            /                   \                   10*\/ pi *Gamma(5/4)*Gamma(7/4)                                       25*\/ pi *Gamma(7/4)*Gamma(9/4)                   /                            \          \       5*\/ pi       /    \       5*\/ pi       /         /
- ------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------
                                5                                                                                                                                    5                                                                                                                                                                                    5                                                                                                                                5
210π(−10sin⁡(45)Γ(14)Γ(34)2F3(14,3412,54,74|−425)10πΓ(54)Γ(74)+210cos⁡(45)Γ(34)Γ(54)2F3(34,5432,74,94|−425)25πΓ(74)Γ(94))5−210π(−10(4−5π2)32sin⁡(45)Γ(14)Γ(34)2F3(14,3412,54,74|−(4−5π2)2100)80πΓ(54)Γ(74)+10(4−5π2)52cos⁡(45)Γ(34)Γ(54)2F3(34,5432,74,94|−(4−5π2)2100)400πΓ(74)Γ(94))5−410π(−sin⁡(45)C(2105π)+cos⁡(45)S(2105π))5+10π(4−5π2)(−sin⁡(45)C(104−5π25π)+cos⁡(45)S(104−5π25π))5\frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \sin{\left(\frac{4}{5} \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| {- \frac{4}{25}} \right)}}{10 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{2 \sqrt{10} \cos{\left(\frac{4}{5} \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| {- \frac{4}{25}} \right)}}{25 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{3}{2}} \sin{\left(\frac{4}{5} \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| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{80 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{5}{2}} \cos{\left(\frac{4}{5} \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| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{400 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{4 \sqrt{10} \sqrt{\pi} \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right)\right)}{5} + \frac{\sqrt{10} \sqrt{\pi} \left(4 - \frac{5 \pi}{2}\right) \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right)\right)}{5} 
=
= 
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                                                                                  |                                                  /              |            2 \                                                                     /              |            2 \|                                                                                                                                                                                                                                                                   
                                                                                  |                   3/2                         _  |              |  /    5*pi\  |                             5/2                                  _  |              |  /    5*pi\  ||                                                                                                                                                                                                                                                                   
                                                                                  |    ____ /    5*pi\                           |_  |   1/4, 3/4   | -|4 - ----|  |              ____ /    5*pi\                                    |_  |   3/4, 5/4   | -|4 - ----|  ||                   /                                 _                                                                               _                         \                            /          /           __________\    /           __________\         \
                                                                                  |  \/ 10 *|4 - ----|   *Gamma(1/4)*Gamma(3/4)* |   |              |  \     2  /  |*sin(4/5)   \/ 10 *|4 - ----|   *cos(4/5)*Gamma(3/4)*Gamma(5/4)* |   |              |  \     2  /  ||                   |    ____                        |_  /   1/4, 3/4   |      \                ____                                 |_  /   3/4, 5/4   |      \|                            |          |  ____    /     5*pi |    |  ____    /     5*pi |         |
                  /          /    ____\    /    ____\         \                   |         \     2  /                          2  3 |1/2, 5/4, 7/4 | -------------|                   \     2  /                                   2  3 |3/2, 7/4, 9/4 | -------------||                   |  \/ 10 *Gamma(1/4)*Gamma(3/4)* |   |              | -4/25|*sin(4/5)   2*\/ 10 *cos(4/5)*Gamma(3/4)*Gamma(5/4)* |   |              | -4/25||                            |          |\/ 10 *  /  4 - ---- |    |\/ 10 *  /  4 - ---- |         |
      ____   ____ |          |2*\/ 10 |    |2*\/ 10 |         |       ____   ____ |                                                  \              |      100     /                                                                     \              |      100     /|       ____   ____ |                               2  3 \1/2, 5/4, 7/4 |      /                                                    2  3 \3/2, 7/4, 9/4 |      /|     ____   ____ /    5*pi\ |          |       \/        2   |    |       \/        2   |         |
  4*\/ 10 *\/ pi *|cos(4/5)*S|--------| - C|--------|*sin(4/5)|   2*\/ 10 *\/ pi *|- ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------|   2*\/ 10 *\/ pi *|- ------------------------------------------------------------------ + --------------------------------------------------------------------|   \/ 10 *\/ pi *|4 - ----|*|cos(4/5)*S|---------------------| - C|---------------------|*sin(4/5)|
                  |          |    ____|    |    ____|         |                   |                                   ____                                                                                        ____                                                  |                   |                        ____                                                                  ____                                         |                 \     2  / |          |           ____      |    |           ____      |         |
                  \          \5*\/ pi /    \5*\/ pi /         /                   \                              80*\/ pi *Gamma(5/4)*Gamma(7/4)                                                            400*\/ pi *Gamma(7/4)*Gamma(9/4)                            /                   \                   10*\/ pi *Gamma(5/4)*Gamma(7/4)                                       25*\/ pi *Gamma(7/4)*Gamma(9/4)                   /                            \          \       5*\/ pi       /    \       5*\/ pi       /         /
- ------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------
                                5                                                                                                                                    5                                                                                                                                                                                    5                                                                                                                                5
210π(−10sin⁡(45)Γ(14)Γ(34)2F3(14,3412,54,74|−425)10πΓ(54)Γ(74)+210cos⁡(45)Γ(34)Γ(54)2F3(34,5432,74,94|−425)25πΓ(74)Γ(94))5−210π(−10(4−5π2)32sin⁡(45)Γ(14)Γ(34)2F3(14,3412,54,74|−(4−5π2)2100)80πΓ(54)Γ(74)+10(4−5π2)52cos⁡(45)Γ(34)Γ(54)2F3(34,5432,74,94|−(4−5π2)2100)400πΓ(74)Γ(94))5−410π(−sin⁡(45)C(2105π)+cos⁡(45)S(2105π))5+10π(4−5π2)(−sin⁡(45)C(104−5π25π)+cos⁡(45)S(104−5π25π))5\frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \sin{\left(\frac{4}{5} \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| {- \frac{4}{25}} \right)}}{10 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{2 \sqrt{10} \cos{\left(\frac{4}{5} \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| {- \frac{4}{25}} \right)}}{25 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{2 \sqrt{10} \sqrt{\pi} \left(- \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{3}{2}} \sin{\left(\frac{4}{5} \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| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{80 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{10} \left(4 - \frac{5 \pi}{2}\right)^{\frac{5}{2}} \cos{\left(\frac{4}{5} \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| {- \frac{\left(4 - \frac{5 \pi}{2}\right)^{2}}{100}} \right)}}{400 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{5} - \frac{4 \sqrt{10} \sqrt{\pi} \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{2 \sqrt{10}}{5 \sqrt{\pi}}\right)\right)}{5} + \frac{\sqrt{10} \sqrt{\pi} \left(4 - \frac{5 \pi}{2}\right) \left(- \sin{\left(\frac{4}{5} \right)} C\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right) + \cos{\left(\frac{4}{5} \right)} S\left(\frac{\sqrt{10} \sqrt{4 - \frac{5 \pi}{2}}}{5 \sqrt{\pi}}\right)\right)}{5} 
-4*sqrt(10)*sqrt(pi)*(cos(4/5)*fresnels(2*sqrt(10)/(5*sqrt(pi))) - fresnelc(2*sqrt(10)/(5*sqrt(pi)))*sin(4/5))/5 - 2*sqrt(10)*sqrt(pi)*(-sqrt(10)*(4 - 5*pi/2)^(3/2)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -(4 - 5*pi/2)^2/100)*sin(4/5)/(80*sqrt(pi)*gamma(5/4)*gamma(7/4)) + sqrt(10)*(4 - 5*pi/2)^(5/2)*cos(4/5)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -(4 - 5*pi/2)^2/100)/(400*sqrt(pi)*gamma(7/4)*gamma(9/4)))/5 + 2*sqrt(10)*sqrt(pi)*(-sqrt(10)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -4/25)*sin(4/5)/(10*sqrt(pi)*gamma(5/4)*gamma(7/4)) + 2*sqrt(10)*cos(4/5)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -4/25)/(25*sqrt(pi)*gamma(7/4)*gamma(9/4)))/5 + sqrt(10)*sqrt(pi)*(4 - 5*pi/2)*(cos(4/5)*fresnels(sqrt(10)*sqrt(4 - 5*pi/2)/(5*sqrt(pi))) - fresnelc(sqrt(10)*sqrt(4 - 5*pi/2)/(5*sqrt(pi)))*sin(4/5))/5
Respuesta numérica [src] 
(0.327486816197669 + 0.9423047673792j)
(0.327486816197669 + 0.9423047673792j)

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

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