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Resolución de integrales/ cos(x+n/8)^2

Integral de cos(x+n/8)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  n               
  -               
  4               
  /               
 |                
 |     2/    n\   
 |  cos |x + -| dx
 |      \    8/   
 |                
/                 
0
∫0n4cos⁡2(n8+x) dx\int\limits_{0}^{\frac{n}{4}} \cos^{2}{\left(\frac{n}{8} + x \right)}\, dx 
Integral(cos(x + n/8)^2, (x, 0, n/4))
Respuesta (Indefinida) [src] 
  /                                                                           3/x   n \                             /x   n \                              4/x   n \                              2/x   n \         
 |                                                                       2*tan |- + --|                        2*tan|- + --|                         x*tan |- + --|                       2*x*tan |- + --|         
 |    2/    n\                           x                                     \2   16/                             \2   16/                               \2   16/                               \2   16/         
 | cos |x + -| dx = C + ----------------------------------- - ----------------------------------- + ----------------------------------- + ----------------------------------- + -----------------------------------
 |     \    8/                   4/x   n \        2/x   n \            4/x   n \        2/x   n \            4/x   n \        2/x   n \            4/x   n \        2/x   n \            4/x   n \        2/x   n \
 |                      2 + 2*tan |- + --| + 4*tan |- + --|   2 + 2*tan |- + --| + 4*tan |- + --|   2 + 2*tan |- + --| + 4*tan |- + --|   2 + 2*tan |- + --| + 4*tan |- + --|   2 + 2*tan |- + --| + 4*tan |- + --|
/                                 \2   16/         \2   16/             \2   16/         \2   16/             \2   16/         \2   16/             \2   16/         \2   16/             \2   16/         \2   16/
∫cos⁡2(n8+x) dx=C+xtan⁡4(n16+x2)2tan⁡4(n16+x2)+4tan⁡2(n16+x2)+2+2xtan⁡2(n16+x2)2tan⁡4(n16+x2)+4tan⁡2(n16+x2)+2+x2tan⁡4(n16+x2)+4tan⁡2(n16+x2)+2−2tan⁡3(n16+x2)2tan⁡4(n16+x2)+4tan⁡2(n16+x2)+2+2tan⁡(n16+x2)2tan⁡4(n16+x2)+4tan⁡2(n16+x2)+2\int \cos^{2}{\left(\frac{n}{8} + x \right)}\, dx = C + \frac{x \tan^{4}{\left(\frac{n}{16} + \frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{n}{16} + \frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 2} - \frac{2 \tan^{3}{\left(\frac{n}{16} + \frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 2} + \frac{2 \tan{\left(\frac{n}{16} + \frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{n}{16} + \frac{x}{2} \right)} + 2}
Simplificar
Respuesta [src] 
   /3*n\    /3*n\      /n\    /n\        2/3*n\        2/3*n\
cos|---|*sin|---|   cos|-|*sin|-|   n*cos |---|   n*sin |---|
   \ 8 /    \ 8 /      \8/    \8/         \ 8 /         \ 8 /
----------------- - ------------- + ----------- + -----------
        2                 2              8             8
nsin⁡2(3n8)8+ncos⁡2(3n8)8−sin⁡(n8)cos⁡(n8)2+sin⁡(3n8)cos⁡(3n8)2\frac{n \sin^{2}{\left(\frac{3 n}{8} \right)}}{8} + \frac{n \cos^{2}{\left(\frac{3 n}{8} \right)}}{8} - \frac{\sin{\left(\frac{n}{8} \right)} \cos{\left(\frac{n}{8} \right)}}{2} + \frac{\sin{\left(\frac{3 n}{8} \right)} \cos{\left(\frac{3 n}{8} \right)}}{2} 
=
= 
   /3*n\    /3*n\      /n\    /n\        2/3*n\        2/3*n\
cos|---|*sin|---|   cos|-|*sin|-|   n*cos |---|   n*sin |---|
   \ 8 /    \ 8 /      \8/    \8/         \ 8 /         \ 8 /
----------------- - ------------- + ----------- + -----------
        2                 2              8             8
nsin⁡2(3n8)8+ncos⁡2(3n8)8−sin⁡(n8)cos⁡(n8)2+sin⁡(3n8)cos⁡(3n8)2\frac{n \sin^{2}{\left(\frac{3 n}{8} \right)}}{8} + \frac{n \cos^{2}{\left(\frac{3 n}{8} \right)}}{8} - \frac{\sin{\left(\frac{n}{8} \right)} \cos{\left(\frac{n}{8} \right)}}{2} + \frac{\sin{\left(\frac{3 n}{8} \right)} \cos{\left(\frac{3 n}{8} \right)}}{2} 
cos(3*n/8)*sin(3*n/8)/2 - cos(n/8)*sin(n/8)/2 + n*cos(3*n/8)^2/8 + n*sin(3*n/8)^2/8

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

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