Solution
Rewrite the sine using the power reducing formula sin4(α)=−2cos(2α)+8cos(4α)+83 with α=x:
∫sin4(x)dx=∫(−2cos(2x)+8cos(4x)+83)dx
Apply the constant multiple rule ∫cf(x)dx=c∫f(x)dx with c=81 and f(x)=−4cos(2x)+cos(4x)+3:
∫(−2cos(2x)+8cos(4x)+83)dx=(8∫(−4cos(2x)+cos(4x)+3)dx)
Integrate term by term:
8∫(−4cos(2x)+cos(4x)+3)dx=8(∫3dx−∫4cos(2x)dx+∫cos(4x)dx)
Apply the constant rule ∫cdx=cx with c=3:
−8∫4cos(2x)dx+8∫cos(4x)dx+8∫3dx=−8∫4cos(2x)dx+8∫cos(4x)dx+8(3x)
Apply the constant multiple rule ∫cf(x)dx=c∫f(x)dx with c=4 and f(x)=cos(2x):
83x+8∫cos(4x)dx−8∫4cos(2x)dx=83x+8∫cos(4x)dx−8(4∫cos(2x)dx)
Let u=2x.
Then du=(2x)′dx=2dx (steps can be seen »), and we have that dx=2du.
C'est pourquoi,
83x+8∫cos(4x)dx−2∫cos(2x)dx=83x+8∫cos(4x)dx−2∫2cos(u)du
Apply the constant multiple rule ∫cf(u)du=c∫f(u)du with c=21 and f(u)=cos(u):
83x+8∫cos(4x)dx−2∫2cos(u)du=83x+8∫cos(4x)dx−2(2∫cos(u)du)
The integral of the cosine is ∫cos(u)du=sin(u):
83x+8∫cos(4x)dx−4∫cos(u)du=83x+8∫cos(4x)dx−4sin(u)
Recall that u=2x:
83x+8∫cos(4x)dx−4sin(u)=83x+8∫cos(4x)dx−4sin((2x))
Let u=4x.
Then du=(4x)′dx=4dx (steps can be seen »), and we have that dx=4du.
The integral becomes
83x−4sin(2x)+8∫cos(4x)dx=83x−4sin(2x)+8∫4cos(u)du
Apply the constant multiple rule ∫cf(u)du=c∫f(u)du with c=41 and f(u)=cos(u):
83x−4sin(2x)+8∫4cos(u)du=83x−4sin(2x)+8(4∫cos(u)du)
The integral of the cosine is ∫cos(u)du=sin(u):
83x−4sin(2x)+32∫cos(u)du=83x−4sin(2x)+32sin(u)
Recall that u=4x:
83x−4sin(2x)+32sin(u)=83x−4sin(2x)+32sin((4x))
C'est pourquoi,
∫sin4(x)dx=83x−4sin(2x)+32sin(4x)
Simplifier :
∫sin4(x)dx=3212x−8sin(2x)+sin(4x)
Ajouter la constante d'intégration :
∫sin4(x)dx=3212x−8sin(2x)+sin(4x)+C
Answer: ∫sin4(x)dx=3212x−8sin(2x)+sin(4x)+C