Sr Examen
Integral de cos(2.2*x)*cos((x-2.3)u) dx
Solución
Ha introducido
[src]
37 -- 10 / | | /11*x\ // 23\ \ | cos|----|*cos||x - --|*u| dx | \ 5 / \\ 10/ / | / 23 -- 10
$$\int\limits_{\frac{23}{10}}^{\frac{37}{10}} \cos{\left(\frac{11 x}{5} \right)} \cos{\left(u \left(x - \frac{23}{10}\right) \right)}\, dx$$
Integral(cos(11*x/5)*cos((x - 23/10)*u), (x, 23/10, 37/10))
Respuesta (Indefinida)
[src]
// 2/11*x\ \
|| -5*sin |----| |
|| \ 5 / |
// 2/11*x\ 2/11*x\ /11*x\ /11*x\ \ || ------------- for u = -11/5|
||x*cos |----| x*sin |----| 5*cos|----|*sin|----| | || 22 |
|| \ 5 / \ 5 / \ 5 / \ 5 / | || |
/ ||------------ + ------------ + --------------------- for Or(u = -11/5, u = 11/5)| || 2/11*x\ |
| || 2 2 22 | || 5*sin |----| |
| /11*x\ // 23\ \ || | /23*u\ || \ 5 / | /23*u\
| cos|----|*cos||x - --|*u| dx = C + |< /11*x\ /11*x\ |*cos|----| + |< ------------ for u = 11/5 |*sin|----|
| \ 5 / \\ 10/ / || 55*cos(u*x)*sin|----| 25*u*cos|----|*sin(u*x) | \ 10 / || 22 | \ 10 /
| || \ 5 / \ 5 / | || |
/ || - --------------------- + ----------------------- otherwise | || /11*x\ /11*x\ |
|| 2 2 | || 55*sin|----|*sin(u*x) 25*u*cos|----|*cos(u*x) |
|| -121 + 25*u -121 + 25*u | || \ 5 / \ 5 / |
\\ / ||- --------------------- - ----------------------- otherwise |
|| 2 2 |
|| -121 + 25*u -121 + 25*u |
\\ /
$$\int \cos{\left(\frac{11 x}{5} \right)} \cos{\left(u \left(x - \frac{23}{10}\right) \right)}\, dx = C + \left(\begin{cases} \frac{x \sin^{2}{\left(\frac{11 x}{5} \right)}}{2} + \frac{x \cos^{2}{\left(\frac{11 x}{5} \right)}}{2} + \frac{5 \sin{\left(\frac{11 x}{5} \right)} \cos{\left(\frac{11 x}{5} \right)}}{22} & \text{for}\: u = - \frac{11}{5} \vee u = \frac{11}{5} \\\frac{25 u \sin{\left(u x \right)} \cos{\left(\frac{11 x}{5} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{11 x}{5} \right)} \cos{\left(u x \right)}}{25 u^{2} - 121} & \text{otherwise} \end{cases}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\begin{cases} - \frac{5 \sin^{2}{\left(\frac{11 x}{5} \right)}}{22} & \text{for}\: u = - \frac{11}{5} \\\frac{5 \sin^{2}{\left(\frac{11 x}{5} \right)}}{22} & \text{for}\: u = \frac{11}{5} \\- \frac{25 u \cos{\left(\frac{11 x}{5} \right)} \cos{\left(u x \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{11 x}{5} \right)} \sin{\left(u x \right)}}{25 u^{2} - 121} & \text{otherwise} \end{cases}\right) \sin{\left(\frac{23 u}{10} \right)}$$
Respuesta
[src]
/ / 2/407\ 2/407\ /407\ /407\\ / 2/253\ 2/253\ /253\ /253\\ 2/407\ /23*u\ 2/253\ /23*u\ | |37*cos |---| 37*sin |---| 5*cos|---|*sin|---|| |23*cos |---| 23*sin |---| 5*cos|---|*sin|---|| 5*sin |---|*sin|----| 5*sin |---|*sin|----| | | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ \ 50/ \ 10 / \ 50/ \ 10 / | |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + --------------------- for u = -11/5 | \ 20 20 22 / \ 10 / \ 20 20 22 / \ 10 / 22 22 | | / 2/407\ 2/407\ /407\ /407\\ / 2/253\ 2/253\ /253\ /253\\ 2/253\ /23*u\ 2/407\ /23*u\ | |37*cos |---| 37*sin |---| 5*cos|---|*sin|---|| |23*cos |---| 23*sin |---| 5*cos|---|*sin|---|| 5*sin |---|*sin|----| 5*sin |---|*sin|----| | | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ \ 50/ \ 10 / \ 50/ \ 10 / < |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + --------------------- for u = 11/5 | \ 20 20 22 / \ 10 / \ 20 20 22 / \ 10 / 22 22 | |/ /37*u\ /407\ /407\ /37*u\\ / /407\ /37*u\ /407\ /37*u\\ / /23*u\ /253\ /253\ /23*u\\ / /253\ /23*u\ /253\ /23*u\\ || 55*cos|----|*sin|---| 25*u*cos|---|*sin|----|| | 55*sin|---|*sin|----| 25*u*cos|---|*cos|----|| | 55*cos|----|*sin|---| 25*u*cos|---|*sin|----|| | 55*sin|---|*sin|----| 25*u*cos|---|*cos|----|| || \ 10 / \ 50/ \ 50/ \ 10 /| /23*u\ | \ 50/ \ 10 / \ 50/ \ 10 /| /23*u\ | \ 10 / \ 50/ \ 50/ \ 10 /| /23*u\ | \ 50/ \ 10 / \ 50/ \ 10 /| /23*u\ ||- --------------------- + -----------------------|*cos|----| + |- --------------------- - -----------------------|*sin|----| - |- --------------------- + -----------------------|*cos|----| - |- --------------------- - -----------------------|*sin|----| otherwise || 2 2 | \ 10 / | 2 2 | \ 10 / | 2 2 | \ 10 / | 2 2 | \ 10 / |\ -121 + 25*u -121 + 25*u / \ -121 + 25*u -121 + 25*u / \ -121 + 25*u -121 + 25*u / \ -121 + 25*u -121 + 25*u / \
$$\begin{cases} - \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = - \frac{11}{5} \\- \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = \frac{11}{5} \\- \left(\frac{25 u \sin{\left(\frac{23 u}{10} \right)} \cos{\left(\frac{253}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{25 u \sin{\left(\frac{37 u}{10} \right)} \cos{\left(\frac{407}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} - \left(- \frac{25 u \cos{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} + \left(- \frac{25 u \cos{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \sin{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} & \text{otherwise} \end{cases}$$
=
=
/ / 2/407\ 2/407\ /407\ /407\\ / 2/253\ 2/253\ /253\ /253\\ 2/407\ /23*u\ 2/253\ /23*u\ | |37*cos |---| 37*sin |---| 5*cos|---|*sin|---|| |23*cos |---| 23*sin |---| 5*cos|---|*sin|---|| 5*sin |---|*sin|----| 5*sin |---|*sin|----| | | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ \ 50/ \ 10 / \ 50/ \ 10 / | |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + --------------------- for u = -11/5 | \ 20 20 22 / \ 10 / \ 20 20 22 / \ 10 / 22 22 | | / 2/407\ 2/407\ /407\ /407\\ / 2/253\ 2/253\ /253\ /253\\ 2/253\ /23*u\ 2/407\ /23*u\ | |37*cos |---| 37*sin |---| 5*cos|---|*sin|---|| |23*cos |---| 23*sin |---| 5*cos|---|*sin|---|| 5*sin |---|*sin|----| 5*sin |---|*sin|----| | | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ | \ 50/ \ 50/ \ 50/ \ 50/| /23*u\ \ 50/ \ 10 / \ 50/ \ 10 / < |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + --------------------- for u = 11/5 | \ 20 20 22 / \ 10 / \ 20 20 22 / \ 10 / 22 22 | |/ /37*u\ /407\ /407\ /37*u\\ / /407\ /37*u\ /407\ /37*u\\ / /23*u\ /253\ /253\ /23*u\\ / /253\ /23*u\ /253\ /23*u\\ || 55*cos|----|*sin|---| 25*u*cos|---|*sin|----|| | 55*sin|---|*sin|----| 25*u*cos|---|*cos|----|| | 55*cos|----|*sin|---| 25*u*cos|---|*sin|----|| | 55*sin|---|*sin|----| 25*u*cos|---|*cos|----|| || \ 10 / \ 50/ \ 50/ \ 10 /| /23*u\ | \ 50/ \ 10 / \ 50/ \ 10 /| /23*u\ | \ 10 / \ 50/ \ 50/ \ 10 /| /23*u\ | \ 50/ \ 10 / \ 50/ \ 10 /| /23*u\ ||- --------------------- + -----------------------|*cos|----| + |- --------------------- - -----------------------|*sin|----| - |- --------------------- + -----------------------|*cos|----| - |- --------------------- - -----------------------|*sin|----| otherwise || 2 2 | \ 10 / | 2 2 | \ 10 / | 2 2 | \ 10 / | 2 2 | \ 10 / |\ -121 + 25*u -121 + 25*u / \ -121 + 25*u -121 + 25*u / \ -121 + 25*u -121 + 25*u / \ -121 + 25*u -121 + 25*u / \
$$\begin{cases} - \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = - \frac{11}{5} \\- \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = \frac{11}{5} \\- \left(\frac{25 u \sin{\left(\frac{23 u}{10} \right)} \cos{\left(\frac{253}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{25 u \sin{\left(\frac{37 u}{10} \right)} \cos{\left(\frac{407}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} - \left(- \frac{25 u \cos{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} + \left(- \frac{25 u \cos{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \sin{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} & \text{otherwise} \end{cases}$$
Piecewise(((37*cos(407/50)^2/20 + 37*sin(407/50)^2/20 + 5*cos(407/50)*sin(407/50)/22)*cos(23*u/10) - (23*cos(253/50)^2/20 + 23*sin(253/50)^2/20 + 5*cos(253/50)*sin(253/50)/22)*cos(23*u/10) - 5*sin(407/50)^2*sin(23*u/10)/22 + 5*sin(253/50)^2*sin(23*u/10)/22, u = -11/5), ((37*cos(407/50)^2/20 + 37*sin(407/50)^2/20 + 5*cos(407/50)*sin(407/50)/22)*cos(23*u/10) - (23*cos(253/50)^2/20 + 23*sin(253/50)^2/20 + 5*cos(253/50)*sin(253/50)/22)*cos(23*u/10) - 5*sin(253/50)^2*sin(23*u/10)/22 + 5*sin(407/50)^2*sin(23*u/10)/22, u = 11/5), ((-55*cos(37*u/10)*sin(407/50)/(-121 + 25*u^2) + 25*u*cos(407/50)*sin(37*u/10)/(-121 + 25*u^2))*cos(23*u/10) + (-55*sin(407/50)*sin(37*u/10)/(-121 + 25*u^2) - 25*u*cos(407/50)*cos(37*u/10)/(-121 + 25*u^2))*sin(23*u/10) - (-55*cos(23*u/10)*sin(253/50)/(-121 + 25*u^2) + 25*u*cos(253/50)*sin(23*u/10)/(-121 + 25*u^2))*cos(23*u/10) - (-55*sin(253/50)*sin(23*u/10)/(-121 + 25*u^2) - 25*u*cos(253/50)*cos(23*u/10)/(-121 + 25*u^2))*sin(23*u/10), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.