How do you find the antiderivative of #cos^4 x dx#?

Question

Ezekiel Alexander · Accepted Answer

to find this integration or anti-derivative we use Reduction formula

Reduction formula #intcos^n xdx=(cos^(n-1) xsinx)/n +(n-1)/n intcos^(n-2) x dx# using n=4 we get #intcos^4 xdx=(cos^(4-1) xsinx)/4 +(4-1)/4 intcos^(4-2) x dx# #intcos^4 xdx=(cos^(3) xsinx)/4 +(3)/4 intcos^(2) x dx# #intcos^4 xdx=(cos^(3) xsinx)/4 +(3)/4 int(1/2 cos2x+1/2)dx)# #intcos^4 xdx=(cos^(3) xsinx)/4 +(3)/8 int(cos2x)dx+3/8 int1dx)# #intcos^4 xdx=(cos^(3) xsinx)/4 +(3)/8 sinx cosx+3/8 x# #intcos^4 xdx=1/32 (12x+8sin(2x)+sin4x)+C# that is the right answer.

Elijah Abram · Answer

# 1/32{12x+8sin2x+sin4x}+C.# Recall that, #cos2x=2cos^2x-1," so that, "cos^2x=(1+cos2x)/2.# #:. cos^4x=((1+cos2x)/2)^2,# #=1/4{1+2cos2x+cos^2(2x)},# #=1/4{1+2cos2x+(1+cos4x)/2},# #=1/8(3+4cos2x+cos4x).# # rArr intcos^4xdx=1/8int(3+4cos2x+cos4x)dx,# #=1/8{3x+4*sin(2x)/2+sin(4x)/4},# #=1/32{12x+8sin2x+sin4x}+C.#

Emma Aldridge · Answer

undefined To find the antiderivative of cos^4(x) dx, you can use the power-reducing formula for cosine, which states that cos^2(x) = (1 + cos(2x))/2. By applying this formula twice, you can reduce cos^4(x) to a combination of cos(2x) terms. Then, integrate each term separately using the power rule for integration.