### Video Transcript

Use integration by parts to find
the exact value of the integral of π₯ squared sin two π₯ ππ₯ between zero and π
by four.

Integration by parts uses the
formula that the integral of π’π£ dash is equal to π’π£ minus the integral of π£π’
dash, where π’ dash is the differential of π’, and π£ dash is the differential of
π£. Our first step is to split our
initial expression π₯ squared sin two π₯ into π’ and π£ dash. We will let π’ equal π₯ squared and
π£ dash equals sin two π₯.

To work out an expression for π’
dash, we need to differentiate π₯ squared. π₯ squared differentiated is two
π₯. To work out an expression for π£,
we need to integrate sin two π₯, as integration is the opposite of
differentiation. The integral of sin two π₯ is equal
to negative cos two π₯ over two. Multiplying π₯ squared by negative
cos two π₯ over two gives us negative π₯ squared cos two π₯ over two. Multiplying two π₯ by negative cos
two π₯ over two gives us negative two π₯ cos two π₯ over two.

As we have two negative signs, this
could be simplified to negative π₯ squared cos two π₯ over two plus the integral of
two π₯ cos two π₯ divided by two. We can also cancel the twos after
the integration sign, by dividing the numerator and denominator by two. We now need to try and integrate π₯
cos two π₯. We can integrate this expression
using parts once again.

We will let π’ equal π₯ and π£ dash
equal cos two π₯. Differentiating π₯ gives us
one. Therefore, π’ dash is equal to
one. Integrating cos two π₯ gives us sin
two π₯ over two. Multiplying π’ and π£, π₯ and sin
two π₯ over two, give us π₯ sin two π₯ over two. Multiplying π’ dash and π£ gives us
sin two π₯ over two. We are now left with negative π₯
squared cos two π₯ over two plus π₯ sin two π₯ over two minus the integral of sin
two π₯ over two.

Integrating the third term, sin two
π₯ over two, gives us negative cos two π₯ over four. Once again, our two negative sins
can turn into a positive. This means that the integral of π₯
squared sin two π₯ is equal to negative π₯ squared cos two π₯ over two plus π₯ sin
two π₯ over two plus cos two π₯ over four. Our final step is to substitute our
two limits, π by four and zero, and subtract the two answers.

Firstly, letβs substitute π₯ equals
π by four. Before starting, itβs worth noting
that our trigonometrical functions are cos two π₯ and sin two π₯. Therefore, we need to calculate cos
of two π by four and sin of two π by four. Two π by four radians is the same
as π by two radians. And π by two radians is equal to
90 degrees. We know from our trig graphs that
cos of 90 is equal to zero. Therefore, cos of π by two radians
must also equal zero.

The sin of 90 degrees is equal to
one. Therefore, the sin of π by two
radians is also equal to one. As cos of π by two is equal to
zero, we know that the first and third terms in our expression will be equal to
zero when π₯ is equal to π by four. The only term that will give us a
value is π₯ sin two π₯ over two. As π₯ is equal to π by four and
sin of two π₯ is equal to one, this term gives us π by four multiplied by one
divided by two. This is equal to π by eight, as π
by four divided by two is equal to π by eight. When π₯ is equal to π by four, the
integral of π₯ squared sin two π₯ equals π by eight.

We also need to consider the lower
limit when π₯ is equal to zero. Once again, from our trig graphs,
we know that cos of zero is equal to one and sin of zero is equal to zero. This means that the middle term π₯
sin two π₯ over two will be equal to zero. It is possible that the first and
third term will have nonzero values, as cos of two π₯ is equal to one.

The first term negative π₯ squared
cos two π₯ over two also has an π₯ term. And as π₯ is equal to zero, this
whole term will also equal zero. Therefore, the only nonzero term is
cos of two π₯ over four. We already know that cos two π₯ is
equal to one. Therefore, cos two π₯ over four is
equal to one-quarter. As we have worked out exact values
for the upper and lower limit, we can now say that the integral of π₯ squared sin
two π₯ between zero and π by four is π by eight minus one-quarter.