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Resolución de integrales/ cos(x-pi/3)/ cos(x-pi/3)^2

Integral de cos(x-pi/3)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |     2/    pi\   
 |  cos |x - --| dx
 |      \    3 /   
 |                 
/                  
0
$$\int\limits_{0}^{1} \cos^{2}{\left(x - \frac{\pi}{3} \right)}\, dx$$ 
Integral(cos(x - pi/3)^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                            /x   pi\                              3/x   pi\                             4/x   pi\                              2/x   pi\         
 |                                                                        2*cot|- + --|                         2*cot |- + --|                        x*cot |- + --|                       2*x*cot |- + --|         
 |    2/    pi\                           x                                    \2   3 /                               \2   3 /                              \2   3 /                               \2   3 /         
 | cos |x - --| dx = C + ----------------------------------- - ----------------------------------- + ----------------------------------- + ----------------------------------- + -----------------------------------
 |     \    3 /                   4/x   pi\        2/x   pi\            4/x   pi\        2/x   pi\            4/x   pi\        2/x   pi\            4/x   pi\        2/x   pi\            4/x   pi\        2/x   pi\
 |                       2 + 2*cot |- + --| + 4*cot |- + --|   2 + 2*cot |- + --| + 4*cot |- + --|   2 + 2*cot |- + --| + 4*cot |- + --|   2 + 2*cot |- + --| + 4*cot |- + --|   2 + 2*cot |- + --| + 4*cot |- + --|
/                                  \2   3 /         \2   3 /             \2   3 /         \2   3 /             \2   3 /         \2   3 /             \2   3 /         \2   3 /             \2   3 /         \2   3 /
$$\int \cos^{2}{\left(x - \frac{\pi}{3} \right)}\, dx = C + \frac{x \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} + \frac{2 x \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} + \frac{x}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} + \frac{2 \cot^{3}{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2} - \frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{3} \right)} + 2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
   2/    pi\      2/    pi\              /    pi\    /    pi\
cos |1 + --|   sin |1 + --|     ___   cos|1 + --|*sin|1 + --|
    \    6 /       \    6 /   \/ 3       \    6 /    \    6 /
------------ + ------------ + ----- - -----------------------
     2              2           8                2
$$- \frac{\sin{\left(\frac{\pi}{6} + 1 \right)} \cos{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\cos^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\sqrt{3}}{8} + \frac{\sin^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2}$$ 
=
= 
   2/    pi\      2/    pi\              /    pi\    /    pi\
cos |1 + --|   sin |1 + --|     ___   cos|1 + --|*sin|1 + --|
    \    6 /       \    6 /   \/ 3       \    6 /    \    6 /
------------ + ------------ + ----- - -----------------------
     2              2           8                2
$$- \frac{\sin{\left(\frac{\pi}{6} + 1 \right)} \cos{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\cos^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2} + \frac{\sqrt{3}}{8} + \frac{\sin^{2}{\left(\frac{\pi}{6} + 1 \right)}}{2}$$ 
cos(1 + pi/6)^2/2 + sin(1 + pi/6)^2/2 + sqrt(3)/8 - cos(1 + pi/6)*sin(1 + pi/6)/2
Respuesta numérica [src] 
0.692942605631488
0.692942605631488

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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