integrate cos^2 (x/3)?

Answer 1

Substitute cos^2u with 1/2 + 1/2cos2u

Then use integration by parts to solve.

For example.

Let u = x/3 du = 1/3dx etc...
v = ? dv = cos^2udu

Answer 2

I will mark integral with S.
Therefore S(cos^2(x/3)dx) = |y=x/3| = 3S(cos^2(y)dy) where | | is substitution.
By partial integration: S(cos^2(y)dy) = sin(y)cos(y) + S(sin^2(y)dy) = sin(y)cos(y) + S((1-cos^2(y))dy)
Follows: 2*S(cos^2(y)dy) = sin(y)cos(y) + y.
Therefore S(cos^2(x/3)dx) = (3/2)*(sin(x/3)cos(x/3) + x/3).