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Resolución de integrales/ (x^2)/ sin(x)/ cos(x^2)*sin(x)

Integral de cos(x^2)*sin(x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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 |  cos\x /*sin(x) dx
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0
$$\int\limits_{0}^{1} \sin{\left(x \right)} \cos{\left(x^{2} \right)}\, dx$$ 
Integral(cos(x^2)*sin(x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                        /          /  ___           \    /  ___           \         \                /          /  ___          \    /  ___          \         \
                             ___   ____ |          |\/ 2 *(-1 + 2*x)|    |\/ 2 *(-1 + 2*x)|         |     ___   ____ |          |\/ 2 *(1 + 2*x)|    |\/ 2 *(1 + 2*x)|         |
  /                        \/ 2 *\/ pi *|cos(1/4)*S|----------------| - C|----------------|*sin(1/4)|   \/ 2 *\/ pi *|cos(1/4)*S|---------------| - C|---------------|*sin(1/4)|
 |                                      |          |        ____    |    |        ____    |         |                |          |        ____   |    |        ____   |         |
 |    / 2\                              \          \    2*\/ pi     /    \    2*\/ pi     /         /                \          \    2*\/ pi    /    \    2*\/ pi    /         /
 | cos\x /*sin(x) dx = C - -------------------------------------------------------------------------- + ------------------------------------------------------------------------
 |                                                             4                                                                           4                                    
/
$$\int \sin{\left(x \right)} \cos{\left(x^{2} \right)}\, dx = C - \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2} \left(2 x - 1\right)}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2} \left(2 x - 1\right)}{2 \sqrt{\pi}}\right)\right)}{4} + \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2} \left(2 x + 1\right)}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2} \left(2 x + 1\right)}{2 \sqrt{\pi}}\right)\right)}{4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
               /          /   ___  \    /   ___  \         \                /          /    ___ \    /    ___ \         \                / /   ___  \                      /   ___  \\
    ___   ____ |          | \/ 2   |    | \/ 2   |         |     ___   ____ |          |3*\/ 2  |    |3*\/ 2  |         |     ___   ____ | | \/ 2   |                      | \/ 2   ||
  \/ 2 *\/ pi *|cos(1/4)*S|--------| - C|--------|*sin(1/4)|   \/ 2 *\/ pi *|cos(1/4)*S|--------| - C|--------|*sin(1/4)|   \/ 2 *\/ pi *|C|--------|*sin(1/4) - cos(1/4)*S|--------||
               |          |    ____|    |    ____|         |                |          |    ____|    |    ____|         |                | |    ____|                      |    ____||
               \          \2*\/ pi /    \2*\/ pi /         /                \          \2*\/ pi /    \2*\/ pi /         /                \ \2*\/ pi /                      \2*\/ pi //
- ---------------------------------------------------------- + ---------------------------------------------------------- + ----------------------------------------------------------
                              2                                                            4                                                            4
$$\frac{\sqrt{2} \sqrt{\pi} \left(- \cos{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right) + \sin{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4} - \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{4} \right)} C\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} S\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4}$$ 
=
= 
               /          /   ___  \    /   ___  \         \                /          /    ___ \    /    ___ \         \                / /   ___  \                      /   ___  \\
    ___   ____ |          | \/ 2   |    | \/ 2   |         |     ___   ____ |          |3*\/ 2  |    |3*\/ 2  |         |     ___   ____ | | \/ 2   |                      | \/ 2   ||
  \/ 2 *\/ pi *|cos(1/4)*S|--------| - C|--------|*sin(1/4)|   \/ 2 *\/ pi *|cos(1/4)*S|--------| - C|--------|*sin(1/4)|   \/ 2 *\/ pi *|C|--------|*sin(1/4) - cos(1/4)*S|--------||
               |          |    ____|    |    ____|         |                |          |    ____|    |    ____|         |                | |    ____|                      |    ____||
               \          \2*\/ pi /    \2*\/ pi /         /                \          \2*\/ pi /    \2*\/ pi /         /                \ \2*\/ pi /                      \2*\/ pi //
- ---------------------------------------------------------- + ---------------------------------------------------------- + ----------------------------------------------------------
                              2                                                            4                                                            4
$$\frac{\sqrt{2} \sqrt{\pi} \left(- \cos{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right) + \sin{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4} - \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{4} \right)} C\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} S\left(\frac{\sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{2} + \frac{\sqrt{2} \sqrt{\pi} \left(- \sin{\left(\frac{1}{4} \right)} C\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right) + \cos{\left(\frac{1}{4} \right)} S\left(\frac{3 \sqrt{2}}{2 \sqrt{\pi}}\right)\right)}{4}$$ 
-sqrt(2)*sqrt(pi)*(cos(1/4)*fresnels(sqrt(2)/(2*sqrt(pi))) - fresnelc(sqrt(2)/(2*sqrt(pi)))*sin(1/4))/2 + sqrt(2)*sqrt(pi)*(cos(1/4)*fresnels(3*sqrt(2)/(2*sqrt(pi))) - fresnelc(3*sqrt(2)/(2*sqrt(pi)))*sin(1/4))/4 + sqrt(2)*sqrt(pi)*(fresnelc(sqrt(2)/(2*sqrt(pi)))*sin(1/4) - cos(1/4)*fresnels(sqrt(2)/(2*sqrt(pi))))/4
Respuesta numérica [src] 
0.389900586124319
0.389900586124319

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

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