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

Integral de sin(x^2+1)*x^2 dx

Solución

Ha introducido [src]

  1                  
  /                  
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 |     / 2    \  2   
 |  sin\x  + 1/*x  dx
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/                    
0

$$\int\limits_{0}^{1} x^{2} \sin{\left(x^{2} + 1 \right)}\, dx$$

Integral(sin(x^2 + 1)*x^2, (x, 0, 1))

Solución detallada

Usamos la integración por partes:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

que $u{\left(x \right)} = x^{2}$ y que $\operatorname{dv}{\left(x \right)} = \sin{\left(x^{2} + 1 \right)}$ .

Entonces $\operatorname{du}{\left(x \right)} = 2 x$ .

Para buscar $v{\left(x \right)}$ :
Ahora resolvemos podintegral.
La integral del producto de una función por una constante es la constante por la integral de esta función:

$\int \sqrt{2} \sqrt{\pi} x \left(\sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)\, dx = \sqrt{2} \sqrt{\pi} \int x \left(\sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)\, dx$
1. Vuelva a escribir el integrando:
  
  $x \left(\sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right) = x \sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + x \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)$
2. Integramos término a término:
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int x \sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\, dx = \sin{\left(1 \right)} \int x C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\frac{\sqrt{2} x^{3} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}$
    Por lo tanto, el resultado es: $\frac{\sqrt{2} x^{3} \sin{\left(1 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}$
  1. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int x \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\, dx = \cos{\left(1 \right)} \int x S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\frac{\sqrt{2} x^{5} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}$
    Por lo tanto, el resultado es: $\frac{\sqrt{2} x^{5} \cos{\left(1 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}$
  El resultado es: $\frac{\sqrt{2} x^{5} \cos{\left(1 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} + \frac{\sqrt{2} x^{3} \sin{\left(1 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}$
Por lo tanto, el resultado es: $\sqrt{2} \sqrt{\pi} \left(\frac{\sqrt{2} x^{5} \cos{\left(1 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} + \frac{\sqrt{2} x^{3} \sin{\left(1 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}\right)$
Ahora simplificar:

$- \frac{2 x^{5} \cos{\left(1 \right)} {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{15} - \frac{2 x^{3} \sin{\left(1 \right)} {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{3} + \frac{\sqrt{2} \sqrt{\pi} x^{2} \sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} x^{2} \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}$
Añadimos la constante de integración:

$- \frac{2 x^{5} \cos{\left(1 \right)} {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{15} - \frac{2 x^{3} \sin{\left(1 \right)} {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{3} + \frac{\sqrt{2} \sqrt{\pi} x^{2} \sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} x^{2} \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}+ \mathrm{constant}$

Respuesta:

$- \frac{2 x^{5} \cos{\left(1 \right)} {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{15} - \frac{2 x^{3} \sin{\left(1 \right)} {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{3} + \frac{\sqrt{2} \sqrt{\pi} x^{2} \sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} x^{2} \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

                                        /                                                                                                                                     \                   /        /    ___\    /    ___\       \
                                        |                                 _  /              |   4 \                                                  _  /              |   4 \|     ___   ____  2 |        |x*\/ 2 |    |x*\/ 2 |       |
  /                                     |  ___  3                        |_  |   1/4, 3/4   | -x  |            ___  5                               |_  |   3/4, 5/4   | -x  ||   \/ 2 *\/ pi *x *|cos(1)*S|-------| + C|-------|*sin(1)|
 |                                      |\/ 2 *x *Gamma(1/4)*Gamma(3/4)* |   |              | ----|*sin(1)   \/ 2 *x *cos(1)*Gamma(3/4)*Gamma(5/4)* |   |              | ----||                   |        |   ____|    |   ____|       |
 |    / 2    \  2            ___   ____ |                               2  3 \1/2, 5/4, 7/4 |  4  /                                                2  3 \3/2, 7/4, 9/4 |  4  /|                   \        \ \/ pi /    \ \/ pi /       /
 | sin\x  + 1/*x  dx = C - \/ 2 *\/ pi *|----------------------------------------------------------------- + -----------------------------------------------------------------| + -------------------------------------------------------
 |                                      |                      ____                                                                ____                                       |                              2                           
/                                       \                 16*\/ pi *Gamma(5/4)*Gamma(7/4)                                     16*\/ pi *Gamma(7/4)*Gamma(9/4)                 /

$$\int x^{2} \sin{\left(x^{2} + 1 \right)}\, dx = C + \frac{\sqrt{2} \sqrt{\pi} x^{2} \left(\sin{\left(1 \right)} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + \cos{\left(1 \right)} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)\right)}{2} - \sqrt{2} \sqrt{\pi} \left(\frac{\sqrt{2} x^{5} \cos{\left(1 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} + \frac{\sqrt{2} x^{3} \sin{\left(1 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{x^{4}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}\right)$$

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

             /        /  ___ \    /  ___ \       \                                                                                                                                                 
  ___   ____ |        |\/ 2  |    |\/ 2  |       |                /                              _                                                                       _                        \
\/ 2 *\/ pi *|cos(1)*S|------| + C|------|*sin(1)|                |  ___                        |_  /   1/4, 3/4   |     \            ___                               |_  /   3/4, 5/4   |     \|
             |        |  ____|    |  ____|       |                |\/ 2 *Gamma(1/4)*Gamma(3/4)* |   |              | -1/4|*sin(1)   \/ 2 *cos(1)*Gamma(3/4)*Gamma(5/4)* |   |              | -1/4||
             \        \\/ pi /    \\/ pi /       /     ___   ____ |                            2  3 \1/2, 5/4, 7/4 |     /                                             2  3 \3/2, 7/4, 9/4 |     /|
-------------------------------------------------- - \/ 2 *\/ pi *|-------------------------------------------------------------- + --------------------------------------------------------------|
                        2                                         |                    ____                                                             ____                                      |
                                                                  \               16*\/ pi *Gamma(5/4)*Gamma(7/4)                                  16*\/ pi *Gamma(7/4)*Gamma(9/4)                /

$$- \sqrt{2} \sqrt{\pi} \left(\frac{\sqrt{2} \sin{\left(1 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} \cos{\left(1 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right) + \frac{\sqrt{2} \sqrt{\pi} \left(\cos{\left(1 \right)} S\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right) + \sin{\left(1 \right)} C\left(\frac{\sqrt{2}}{\sqrt{\pi}}\right)\right)}{2}$$

             /        /  ___ \    /  ___ \       \                                                                                                                                                 
  ___   ____ |        |\/ 2  |    |\/ 2  |       |                /                              _                                                                       _                        \
\/ 2 *\/ pi *|cos(1)*S|------| + C|------|*sin(1)|                |  ___                        |_  /   1/4, 3/4   |     \            ___                               |_  /   3/4, 5/4   |     \|
             |        |  ____|    |  ____|       |                |\/ 2 *Gamma(1/4)*Gamma(3/4)* |   |              | -1/4|*sin(1)   \/ 2 *cos(1)*Gamma(3/4)*Gamma(5/4)* |   |              | -1/4||
             \        \\/ pi /    \\/ pi /       /     ___   ____ |                            2  3 \1/2, 5/4, 7/4 |     /                                             2  3 \3/2, 7/4, 9/4 |     /|
-------------------------------------------------- - \/ 2 *\/ pi *|-------------------------------------------------------------- + --------------------------------------------------------------|
                        2                                         |                    ____                                                             ____                                      |
                                                                  \               16*\/ pi *Gamma(5/4)*Gamma(7/4)                                  16*\/ pi *Gamma(7/4)*Gamma(9/4)                /

sqrt(2)*sqrt(pi)*(cos(1)*fresnels(sqrt(2)/sqrt(pi)) + fresnelc(sqrt(2)/sqrt(pi))*sin(1))/2 - sqrt(2)*sqrt(pi)*(sqrt(2)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -1/4)*sin(1)/(16*sqrt(pi)*gamma(5/4)*gamma(7/4)) + sqrt(2)*cos(1)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -1/4)/(16*sqrt(pi)*gamma(7/4)*gamma(9/4)))

Respuesta numérica [src]

0.321890797296217

0.321890797296217

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

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Integral de sin(x^2+1)*x^2 dx

Límites de integración:

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Definida a trozos:

Solución