Evaluate I=\int\cos^4(x)dx?
Symbolab says to use integration by parts.
However, this question originates in a "Trigonometric Integrals" packet... so how would I go about solving this?
Symbolab says to use integration by parts.
However, this question originates in a "Trigonometric Integrals" packet... so how would I go about solving this?
1 Answer
Apr 9, 2018
I=1/32(12x+sin(4x)+8sin(2x))+C
Explanation:
We want to solve
I=intcos^4(x)dx
I always looking to reduce the powers of the integrand,
for integrals of this type
You could use the identity
color(blue)(cos^2(x)=1/2(1+cos(2x))
Thus
I=int(cos^2(x))^2dx
color(white)(I)=1/4int(1+cos(2x))^2dx
color(white)(I)=1/4int1+cos^2(2x)+2cos(2x)dx
color(white)(I)=1/4int1+1/2(1+cos(4x))+2cos(2x)dx
color(white)(I)=1/4int3/2+1/2cos(4x)+2cos(2x)dx
Which is much nicer to integrate
I=1/4(3/2x+1/8sin(4x)+sin(2x))+C
color(white)(I)=1/32(12x+sin(4x)+8sin(2x))+C