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