We're asked to evaluate the indefinite integral,
the integral of the fourth power of sine 2x.
Because we have an even power on sine 2x,
we make repeated use of the 1/2 angle identities
or the power-reducing formula that's shown above.
And since sine squared x is equal to 1/2
times the quantity one minus cosine 2x,
the fourth power of sine 2x is equal to the square
of 1/2 times the quantity one minus cosine 4x.
Notice how we double the angle from the left side
of the equation to the right side of the equation.
So again, we have the integral of the square
of 1/2 times the quantity one minus cosine 4x,
and now we need to square this quantity.
1/2 times 1/2 is 1/4,
which gives us the integral of 1/4 times,
and then have two factors of one minus cosine 4x.
And now we need to multiply this out.
We have four products: one, two, three, and four,
which gives us the integral of 1/4 times the quantity,
we have one minus cosine 4x minus cosine 4x,
which is minus two cosine 4x.
And then we have plus cosine squared 4x.
And now we need to make a substitution
for cosine squared 4x using the identity cosine squared x
equals 1/2 times the quantity one minus cosine 2x,
which means cosine squared 4x is equal to 1/2
times the quantity one plus cosine 8x.
And now we need to simplify.
We have 1/4 times the quantity one minus two cosine 4x.
Bring the 1/2 we have plus 1/2 plus 1/2 times cosine 8x.
And now let's close the parentheses by distributing 1/4.
We have the integral of, notice inside the parentheses,
we have one plus 1/2, that's 3/2.
1/4 times 3/2 is 3/8.
And then we have minus 2/4 cosine 4x,
which is minus 1/2 cosine 4x.
And then we have 1/4 times 1/2 cosine 8x,
which gives us plus 1/8 cosine 4x.
And now let's continue on the next slide.
Because we need to perform u substitution
to integrate the second term and third term,
let's write this as three separate integrals.
We have the integral of 3/8 minus 1/2
times the integral of cosine 4x
plus 1/8 times the integral of cosine 8x.
U substitution is not required for the first integral,
but for the second integral, notice u is equal to 4x,
and therefore differential u is equal to 4dx.
Dividing both sides by four,
we have 1/4 du is equal to dx,
which means in terms of u, we have 1/2 times 1/4
times the integral of cosine u du
And then for the third integral we have u equals 8x
and therefore, differential u is equal to 8dx.
Dividing both sides by eight, we have 1/8 du is equal to dx.
So in terms of u, we have 1/8 times 1/8
times the integral of cosine u du.
Now we can go ahead and integrate.
The integral of 3/8 with respect to x is 3/8 x.
And then we have minus 1/8 times the integral of cosine u,
which is equal to 1/8 times sine u,
giving us minus 1/8 sine 4x, since u is equal to 4x.
And then we have plus 1/64 times the integral of cosine u,
which gives us 1/64 sine u, which is 1/64 sine 8x.
And of course, plus C, for the constant of integration.
I hope you found this helpful.