Sr Examen
Integral de (sqrt(x)+sqrt(x+1))sin5x dx
Solución
Ha introducido
[src]
oo / | | / ___ _______\ | \\/ x + \/ x + 1 /*sin(5*x) dx | / 0
$$\int\limits_{0}^{\infty} \left(\sqrt{x} + \sqrt{x + 1}\right) \sin{\left(5 x \right)}\, dx$$
Integral((sqrt(x) + sqrt(x + 1))*sin(5*x), (x, 0, oo))
Respuesta (Indefinida)
[src]
/ \
| _ / | 2\ _ / | 2\|
| ____ 3/2 |_ | 1/4, 3/4 | -25*(1 + x) | ____ 5/2 |_ | 3/4, 5/4 | -25*(1 + x) ||
| \/ 10 *(1 + x) *Gamma(1/4)*Gamma(3/4)* | | | ------------|*sin(5) 5*\/ 10 *(1 + x) *cos(5)*Gamma(3/4)*Gamma(5/4)* | | | ------------|| / ____ ___\ / / ____ _______\ / ____ _______\ \
____ ____ | 2 3 \1/2, 5/4, 7/4 | 4 / 2 3 \3/2, 7/4, 9/4 | 4 /| ____ ____ |\/ 10 *\/ x | ____ ____ | |\/ 10 *\/ 1 + x | |\/ 10 *\/ 1 + x | | _ / | 2\
/ 2*\/ 10 *\/ pi *|- ---------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------| x*\/ 10 *\/ pi *S|------------| \/ 10 *\/ pi *(1 + x)*|cos(5)*S|----------------| - C|----------------|*sin(5)| 5/2 |_ | 3/4, 5/4 | -25*x |
| | ____ ____ | | ____ | | | ____ | | ____ | | 5*x *Gamma(3/4)*Gamma(5/4)* | | | ------|
| / ___ _______\ \ 16*\/ pi *Gamma(5/4)*Gamma(7/4) 16*\/ pi *Gamma(7/4)*Gamma(9/4) / \ \/ pi / \ \ \/ pi / \ \/ pi / / 2 3 \3/2, 7/4, 9/4 | 4 /
| \\/ x + \/ x + 1 /*sin(5*x) dx = C - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------- + ------------------------------------------------------------------------------- - ----------------------------------------------------------
| 5 5 5 4*Gamma(7/4)*Gamma(9/4)
/
$$\int \left(\sqrt{x} + \sqrt{x + 1}\right) \sin{\left(5 x \right)}\, dx = C - \frac{5 x^{\frac{5}{2}} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{25 x^{2}}{4}} \right)}}{4 \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} + \frac{\sqrt{10} \sqrt{\pi} x S\left(\frac{\sqrt{10} \sqrt{x}}{\sqrt{\pi}}\right)}{5} + \frac{\sqrt{10} \sqrt{\pi} \left(x + 1\right) \left(- \sin{\left(5 \right)} C\left(\frac{\sqrt{10} \sqrt{x + 1}}{\sqrt{\pi}}\right) + \cos{\left(5 \right)} S\left(\frac{\sqrt{10} \sqrt{x + 1}}{\sqrt{\pi}}\right)\right)}{5} - \frac{2 \sqrt{10} \sqrt{\pi} \left(\frac{5 \sqrt{10} \left(x + 1\right)^{\frac{5}{2}} \cos{\left(5 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- \frac{25 \left(x + 1\right)^{2}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} - \frac{\sqrt{10} \left(x + 1\right)^{\frac{3}{2}} \sin{\left(5 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- \frac{25 \left(x + 1\right)^{2}}{4}} \right)}}{16 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}\right)}{5}$$
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.