Интеграл x/cos(x)^4 (dx)

Вы ввели

  1           
  /           
 |            
 |     x      
 |  ------- dx
 |     4      
 |  cos (x)   
 |            
/             
0             

$$\int_{0}^{1} \frac{x}{\cos^{4}{\left (x \right )}}\, dx$$

Ответ

  1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
  /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
 |                                      5                                                                                             4                                                                                                                                            2                                          /       2     \                                    3                                              2         /       2     \                  4                                                  4                                              6         /       2     \                  6                                                  6                                          2                                                  2                                              4         /       2     \       
 |     x                           6*tan (1/2)                                     6*tan(1/2)                                    2*tan (1/2)                            2*(pi*I + log(1 - tan(1/2)))                       2*log(1 + tan(1/2))                                2*tan (1/2)                                2*log\1 + tan (1/2)/                               4*tan (1/2)                    2*pi*I          6*tan (1/2)*log\1 + tan (1/2)/             6*tan (1/2)*(pi*I + log(1 - tan(1/2)))             6*tan (1/2)*log(1 + tan(1/2))                  2*tan (1/2)*log\1 + tan (1/2)/             2*tan (1/2)*(pi*I + log(1 - tan(1/2)))             2*tan (1/2)*log(1 + tan(1/2))              6*tan (1/2)*(pi*I + log(1 - tan(1/2)))             6*tan (1/2)*log(1 + tan(1/2))                  6*tan (1/2)*log\1 + tan (1/2)/       
 |  ------- dx = - -------------------------------------------- - -------------------------------------------- - -------------------------------------------- - -------------------------------------------- - -------------------------------------------- + -------------------------------------------- + -------------------------------------------- + -------------------------------------------- - ------ - -------------------------------------------- - -------------------------------------------- - -------------------------------------------- - -------------------------------------------- + -------------------------------------------- + -------------------------------------------- + -------------------------------------------- + -------------------------------------------- + --------------------------------------------
 |     4                     4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2          3                4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2                  4             6             2     
 |  cos (x)        -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)            -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)   -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)
 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       

$$\int_{0}^{1} \frac{x}{\cos^{4}{\left (x \right )}}\, dx = \frac{6 \log{\left (\tan{\left (\frac{1}{2} \right )} + 1 \right )} \tan^{2}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} + \frac{4 \tan^{3}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} + \frac{2 \tan^{2}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} + \frac{2 \log{\left (\tan^{2}{\left (\frac{1}{2} \right )} + 1 \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} + \frac{6 \log{\left (\tan^{2}{\left (\frac{1}{2} \right )} + 1 \right )} \tan^{4}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} + \frac{2 \log{\left (\tan{\left (\frac{1}{2} \right )} + 1 \right )} \tan^{6}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{2 \log{\left (\tan^{2}{\left (\frac{1}{2} \right )} + 1 \right )} \tan^{6}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{2 \tan^{4}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{6 \log{\left (\tan{\left (\frac{1}{2} \right )} + 1 \right )} \tan^{4}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{6 \tan^{5}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{6 \log{\left (\tan^{2}{\left (\frac{1}{2} \right )} + 1 \right )} \tan^{2}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{2 \log{\left (\tan{\left (\frac{1}{2} \right )} + 1 \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{6 \tan{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} + \frac{6 \left(\log{\left (- \tan{\left (\frac{1}{2} \right )} + 1 \right )} + i \pi\right) \tan^{2}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{2 \pi}{3} i + \frac{2 \left(\log{\left (- \tan{\left (\frac{1}{2} \right )} + 1 \right )} + i \pi\right) \tan^{6}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{6 \left(\log{\left (- \tan{\left (\frac{1}{2} \right )} + 1 \right )} + i \pi\right) \tan^{4}{\left (\frac{1}{2} \right )}}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}} - \frac{2 \log{\left (- \tan{\left (\frac{1}{2} \right )} + 1 \right )} + 2 i \pi}{-3 - 9 \tan^{4}{\left (\frac{1}{2} \right )} + 3 \tan^{6}{\left (\frac{1}{2} \right )} + 9 \tan^{2}{\left (\frac{1}{2} \right )}}$$

Численный ответ

2.00191085687268

Ответ (Неопределённый)

  /                                    4/x\                                 /       /x\\                             /        /x\\                                2/x\                                 /       2/x\\                                 5/x\                                      /x\                            2/x\    /       2/x\\                    4/x\    /       /x\\                     4/x\    /        /x\\                    6/x\    /       2/x\\                    6/x\    /       /x\\                     6/x\    /        /x\\                             3/x\                            2/x\    /       /x\\                     2/x\    /        /x\\                    4/x\    /       2/x\\      
 |                                2*tan |-|                            2*log|1 + tan|-||                        2*log|-1 + tan|-||                           2*tan |-|                            2*log|1 + tan |-||                          6*x*tan |-|                               6*x*tan|-|                       6*tan |-|*log|1 + tan |-||               6*tan |-|*log|1 + tan|-||                6*tan |-|*log|-1 + tan|-||               2*tan |-|*log|1 + tan |-||               2*tan |-|*log|1 + tan|-||                2*tan |-|*log|-1 + tan|-||                      4*x*tan |-|                       6*tan |-|*log|1 + tan|-||                6*tan |-|*log|-1 + tan|-||               6*tan |-|*log|1 + tan |-||      
 |    x                                 \2/                                 \       \2//                             \        \2//                                 \2/                                 \        \2//                                  \2/                                      \2/                             \2/    \        \2//                     \2/    \       \2//                      \2/    \        \2//                     \2/    \        \2//                     \2/    \       \2//                      \2/    \        \2//                              \2/                             \2/    \       \2//                      \2/    \        \2//                     \2/    \        \2//      
 | ------- dx = C - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + --------------------------------------
 |    4                       4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\             4/x\        6/x\        2/x\
 | cos (x)          -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|   -3 - 9*tan |-| + 3*tan |-| + 9*tan |-|
 |                             \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/              \2/         \2/         \2/
/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         

$${{\left(\sin ^2\left(6\,x\right)+\left(6\,\sin \left(4\,x\right)+6 \,\sin \left(2\,x\right)\right)\,\sin \left(6\,x\right)+\cos ^2 \left(6\,x\right)+\left(6\,\cos \left(4\,x\right)+6\,\cos \left(2\,x \right)+2\right)\,\cos \left(6\,x\right)+9\,\sin ^2\left(4\,x\right) +18\,\sin \left(2\,x\right)\,\sin \left(4\,x\right)+9\,\cos ^2\left( 4\,x\right)+\left(18\,\cos \left(2\,x\right)+6\right)\,\cos \left(4 \,x\right)+9\,\sin ^2\left(2\,x\right)+9\,\cos ^2\left(2\,x\right)+6 \,\cos \left(2\,x\right)+1\right)\,\log \left(\sin ^2\left(2\,x \right)+\cos ^2\left(2\,x\right)+2\,\cos \left(2\,x\right)+1\right)+ \left(-2\,\sin \left(4\,x\right)-2\,\sin \left(2\,x\right)+12\,x\, \cos \left(2\,x\right)+4\,x\right)\,\sin \left(6\,x\right)+\left(-2 \,\cos \left(4\,x\right)-12\,x\,\sin \left(2\,x\right)-2\,\cos \left(2\,x\right)\right)\,\cos \left(6\,x\right)-6\,\sin ^2\left(4\, x\right)+\left(-12\,\sin \left(2\,x\right)+36\,x\,\cos \left(2\,x \right)+12\,x\right)\,\sin \left(4\,x\right)-6\,\cos ^2\left(4\,x \right)+\left(-36\,x\,\sin \left(2\,x\right)-12\,\cos \left(2\,x \right)-2\right)\,\cos \left(4\,x\right)-6\,\sin ^2\left(2\,x\right) -6\,\cos ^2\left(2\,x\right)-2\,\cos \left(2\,x\right)}\over{3\, \sin ^2\left(6\,x\right)+\left(18\,\sin \left(4\,x\right)+18\,\sin \left(2\,x\right)\right)\,\sin \left(6\,x\right)+3\,\cos ^2\left(6\, x\right)+\left(18\,\cos \left(4\,x\right)+18\,\cos \left(2\,x\right) +6\right)\,\cos \left(6\,x\right)+27\,\sin ^2\left(4\,x\right)+54\, \sin \left(2\,x\right)\,\sin \left(4\,x\right)+27\,\cos ^2\left(4\,x \right)+\left(54\,\cos \left(2\,x\right)+18\right)\,\cos \left(4\,x \right)+27\,\sin ^2\left(2\,x\right)+27\,\cos ^2\left(2\,x\right)+18 \,\cos \left(2\,x\right)+3}}$$