Integral de (cos^5)4x dx

Solución

Ha introducido [src]

 pi               
 --               
 8                
  /               
 |                
 |     5          
 |  cos (x)*4*x dx
 |                
/                 
0

\int\limits_{0}^{\frac{\pi}{8}} x 4 \cos^{5}{\left(x \right)}\, dx

Integral((cos(x)^5*4)*x, (x, 0, pi/8))

Respuesta (Indefinida) [src]

  /                                                                                                                                     
 |                             5              5            4                    3       2                                   2       3   
 |    5                 596*cos (x)   32*x*sin (x)   32*sin (x)*cos(x)   208*cos (x)*sin (x)          4             16*x*cos (x)*sin (x)
 | cos (x)*4*x dx = C + ----------- + ------------ + ----------------- + ------------------- + 4*x*cos (x)*sin(x) + --------------------
 |                          225            15                15                   45                                         3          
/

\int x 4 \cos^{5}{\left(x \right)}\, dx = C + \frac{32 x \sin^{5}{\left(x \right)}}{15} + \frac{16 x \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}}{3} + 4 x \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \frac{32 \sin^{4}{\left(x \right)} \cos{\left(x \right)}}{15} + \frac{208 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}}{45} + \frac{596 \cos^{5}{\left(x \right)}}{225}

Simplificar

Gráfica

Trazar el gráfico f(x)

Respuesta [src]

                       5/2                   5/2                 2      ___________                  3/2                       ___________            2                   3/2            
            /      ___\           /      ___\         /      ___\      /       ___        /      ___\    /      ___\          /       ___  /      ___\         /      ___\    /      ___\
            |1   \/ 2 |           |1   \/ 2 |         |1   \/ 2 |     /  1   \/ 2         |1   \/ 2 |    |1   \/ 2 |         /  1   \/ 2   |1   \/ 2 |         |1   \/ 2 |    |1   \/ 2 |
        596*|- + -----|      4*pi*|- - -----|      32*|- - -----| *  /   - + -----    208*|- + -----|   *|- - -----|   pi*  /   - - ----- *|- + -----|    2*pi*|- - -----|   *|- + -----|
  596       \2     4  /           \2     4  /         \2     4  /  \/    2     4          \2     4  /    \2     4  /      \/    2     4    \2     4  /         \2     4  /    \2     4  /
- --- + ------------------ + ------------------- + -------------------------------- + ------------------------------ + -------------------------------- + -------------------------------
  225          225                    15                          15                                45                                2                                  3

- \frac{596}{225} + \frac{4 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{5}{2}}}{15} + \frac{32 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{15} + \frac{2 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{3}{2}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)}{3} + \frac{\pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{2}}{2} + \frac{208 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{3}{2}}}{45} + \frac{596 \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{5}{2}}}{225}

                       5/2                   5/2                 2      ___________                  3/2                       ___________            2                   3/2            
            /      ___\           /      ___\         /      ___\      /       ___        /      ___\    /      ___\          /       ___  /      ___\         /      ___\    /      ___\
            |1   \/ 2 |           |1   \/ 2 |         |1   \/ 2 |     /  1   \/ 2         |1   \/ 2 |    |1   \/ 2 |         /  1   \/ 2   |1   \/ 2 |         |1   \/ 2 |    |1   \/ 2 |
        596*|- + -----|      4*pi*|- - -----|      32*|- - -----| *  /   - + -----    208*|- + -----|   *|- - -----|   pi*  /   - - ----- *|- + -----|    2*pi*|- - -----|   *|- + -----|
  596       \2     4  /           \2     4  /         \2     4  /  \/    2     4          \2     4  /    \2     4  /      \/    2     4    \2     4  /         \2     4  /    \2     4  /
- --- + ------------------ + ------------------- + -------------------------------- + ------------------------------ + -------------------------------- + -------------------------------
  225          225                    15                          15                                45                                2                                  3

- \frac{596}{225} + \frac{4 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{5}{2}}}{15} + \frac{32 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{15} + \frac{2 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{3}{2}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)}{3} + \frac{\pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{2}}{2} + \frac{208 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{3}{2}}}{45} + \frac{596 \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{5}{2}}}{225}

-596/225 + 596*(1/2 + sqrt(2)/4)^(5/2)/225 + 4*pi*(1/2 - sqrt(2)/4)^(5/2)/15 + 32*(1/2 - sqrt(2)/4)^2*sqrt(1/2 + sqrt(2)/4)/15 + 208*(1/2 + sqrt(2)/4)^(3/2)*(1/2 - sqrt(2)/4)/45 + pi*sqrt(1/2 - sqrt(2)/4)*(1/2 + sqrt(2)/4)^2/2 + 2*pi*(1/2 - sqrt(2)/4)^(3/2)*(1/2 + sqrt(2)/4)/3

Respuesta numérica [src]

0.255141933168901

0.255141933168901

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Integral de (cos^5)4x dx

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Solución