Sr Examen
Integral de (1+x)^(1/2)cos(6x) dx
Solución
Ha introducido
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1 / | | _______ | \/ 1 + x *cos(6*x) dx | / 0
$$\int\limits_{0}^{1} \sqrt{x + 1} \cos{\left(6 x \right)}\, dx$$
Integral(sqrt(1 + x)*cos(6*x), (x, 0, 1))
Respuesta (Indefinida)
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/ _ _ \
| ___ 3/2 |_ / 1/4, 3/4 | 2\ ___ 5/2 |_ / 3/4, 5/4 | 2\ |
|\/ 3 *(1 + x) *cos(6)*Gamma(1/4)*Gamma(3/4)* | | | -9*(1 + x) | 3*\/ 3 *(1 + x) *Gamma(3/4)*Gamma(5/4)* | | | -9*(1 + x) |*sin(6)| / / ___ _______\ / ___ _______\ \
___ ____ | 2 3 \1/2, 5/4, 7/4 | / 2 3 \3/2, 7/4, 9/4 | / | ___ ____ | |2*\/ 3 *\/ 1 + x | |2*\/ 3 *\/ 1 + x | |
/ 2*\/ 3 *\/ pi *|-------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------| \/ 3 *\/ pi *(1 + x)*|cos(6)*C|-----------------| + S|-----------------|*sin(6)|
| | ____ ____ | | | ____ | | ____ | |
| _______ \ 8*\/ pi *Gamma(5/4)*Gamma(7/4) 4*\/ pi *Gamma(7/4)*Gamma(9/4) / \ \ \/ pi / \ \/ pi / /
| \/ 1 + x *cos(6*x) dx = C - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------
| 3 3
/
$$\int \sqrt{x + 1} \cos{\left(6 x \right)}\, dx = C + \frac{\sqrt{3} \sqrt{\pi} \left(x + 1\right) \left(\cos{\left(6 \right)} C\left(\frac{2 \sqrt{3} \sqrt{x + 1}}{\sqrt{\pi}}\right) + \sin{\left(6 \right)} S\left(\frac{2 \sqrt{3} \sqrt{x + 1}}{\sqrt{\pi}}\right)\right)}{3} - \frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{3 \sqrt{3} \left(x + 1\right)^{\frac{5}{2}} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {- 9 \left(x + 1\right)^{2}} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)} + \frac{\sqrt{3} \left(x + 1\right)^{\frac{3}{2}} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {- 9 \left(x + 1\right)^{2}} \right)}}{8 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}\right)}{3}$$
Gráfica
Respuesta
[src]
/ _ _ \ / _ _ \
| ___ |_ / 3/4, 5/4 | \ ___ |_ / 1/4, 3/4 | \| | ___ |_ / 1/4, 3/4 | \ ___ |_ / 3/4, 5/4 | \ |
|3*\/ 6 *Gamma(3/4)*Gamma(5/4)* | | | -36|*sin(6) \/ 6 *cos(6)*Gamma(1/4)*Gamma(3/4)* | | | -36|| / / ___\ / ___\ \ / / ___\ / ___\ \ |\/ 3 *cos(6)*Gamma(1/4)*Gamma(3/4)* | | | -9| 3*\/ 3 *Gamma(3/4)*Gamma(5/4)* | | | -9|*sin(6)|
___ ____ | 2 3 \3/2, 7/4, 9/4 | / 2 3 \1/2, 5/4, 7/4 | /| ___ ____ | |2*\/ 3 | |2*\/ 3 | | ___ ____ | |2*\/ 6 | |2*\/ 6 | | ___ ____ | 2 3 \1/2, 5/4, 7/4 | / 2 3 \3/2, 7/4, 9/4 | / |
2*\/ 3 *\/ pi *|--------------------------------------------------------------- + -------------------------------------------------------------| \/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)| 2*\/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)| 2*\/ 3 *\/ pi *|------------------------------------------------------------ + --------------------------------------------------------------|
| ____ ____ | | | ____| | ____| | | | ____| | ____| | | ____ ____ |
\ \/ pi *Gamma(7/4)*Gamma(9/4) 4*\/ pi *Gamma(5/4)*Gamma(7/4) / \ \ \/ pi / \ \/ pi / / \ \ \/ pi / \ \/ pi / / \ 8*\/ pi *Gamma(5/4)*Gamma(7/4) 4*\/ pi *Gamma(7/4)*Gamma(9/4) /
- ------------------------------------------------------------------------------------------------------------------------------------------------ - ---------------------------------------------------- + ------------------------------------------------------ + ----------------------------------------------------------------------------------------------------------------------------------------------
3 3 3 3
$$\frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{3} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-9} \right)}}{8 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{3} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-9} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{6} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-36} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{6} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-36} \right)}}{\sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{\sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right)\right)}{3} + \frac{2 \sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right)\right)}{3}$$
=
=
/ _ _ \ / _ _ \
| ___ |_ / 3/4, 5/4 | \ ___ |_ / 1/4, 3/4 | \| | ___ |_ / 1/4, 3/4 | \ ___ |_ / 3/4, 5/4 | \ |
|3*\/ 6 *Gamma(3/4)*Gamma(5/4)* | | | -36|*sin(6) \/ 6 *cos(6)*Gamma(1/4)*Gamma(3/4)* | | | -36|| / / ___\ / ___\ \ / / ___\ / ___\ \ |\/ 3 *cos(6)*Gamma(1/4)*Gamma(3/4)* | | | -9| 3*\/ 3 *Gamma(3/4)*Gamma(5/4)* | | | -9|*sin(6)|
___ ____ | 2 3 \3/2, 7/4, 9/4 | / 2 3 \1/2, 5/4, 7/4 | /| ___ ____ | |2*\/ 3 | |2*\/ 3 | | ___ ____ | |2*\/ 6 | |2*\/ 6 | | ___ ____ | 2 3 \1/2, 5/4, 7/4 | / 2 3 \3/2, 7/4, 9/4 | / |
2*\/ 3 *\/ pi *|--------------------------------------------------------------- + -------------------------------------------------------------| \/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)| 2*\/ 3 *\/ pi *|cos(6)*C|-------| + S|-------|*sin(6)| 2*\/ 3 *\/ pi *|------------------------------------------------------------ + --------------------------------------------------------------|
| ____ ____ | | | ____| | ____| | | | ____| | ____| | | ____ ____ |
\ \/ pi *Gamma(7/4)*Gamma(9/4) 4*\/ pi *Gamma(5/4)*Gamma(7/4) / \ \ \/ pi / \ \/ pi / / \ \ \/ pi / \ \/ pi / / \ 8*\/ pi *Gamma(5/4)*Gamma(7/4) 4*\/ pi *Gamma(7/4)*Gamma(9/4) /
- ------------------------------------------------------------------------------------------------------------------------------------------------ - ---------------------------------------------------- + ------------------------------------------------------ + ----------------------------------------------------------------------------------------------------------------------------------------------
3 3 3 3
$$\frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{3} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-9} \right)}}{8 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{3} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-9} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{2 \sqrt{3} \sqrt{\pi} \left(\frac{\sqrt{6} \cos{\left(6 \right)} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle| {-36} \right)}}{4 \sqrt{\pi} \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{3 \sqrt{6} \sin{\left(6 \right)} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{5}{4}\right) {{}_{2}F_{3}\left(\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle| {-36} \right)}}{\sqrt{\pi} \Gamma\left(\frac{7}{4}\right) \Gamma\left(\frac{9}{4}\right)}\right)}{3} - \frac{\sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{3}}{\sqrt{\pi}}\right)\right)}{3} + \frac{2 \sqrt{3} \sqrt{\pi} \left(\sin{\left(6 \right)} S\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right) + \cos{\left(6 \right)} C\left(\frac{2 \sqrt{6}}{\sqrt{\pi}}\right)\right)}{3}$$
-2*sqrt(3)*sqrt(pi)*(3*sqrt(6)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -36)*sin(6)/(sqrt(pi)*gamma(7/4)*gamma(9/4)) + sqrt(6)*cos(6)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -36)/(4*sqrt(pi)*gamma(5/4)*gamma(7/4)))/3 - sqrt(3)*sqrt(pi)*(cos(6)*fresnelc(2*sqrt(3)/sqrt(pi)) + fresnels(2*sqrt(3)/sqrt(pi))*sin(6))/3 + 2*sqrt(3)*sqrt(pi)*(cos(6)*fresnelc(2*sqrt(6)/sqrt(pi)) + fresnels(2*sqrt(6)/sqrt(pi))*sin(6))/3 + 2*sqrt(3)*sqrt(pi)*(sqrt(3)*cos(6)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -9)/(8*sqrt(pi)*gamma(5/4)*gamma(7/4)) + 3*sqrt(3)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -9)*sin(6)/(4*sqrt(pi)*gamma(7/4)*gamma(9/4)))/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.