Question 1191872
<pre>Let the integral that we wish to integrate be I

{{{I}}}{{{""=""}}}{{{int(sin(x)*sin(4x)dx)}}}

Which factor, if either, is harder to integrate?
I'd say sin(4x) is harder to integrate, so let it be u.

{{{matrix(2,1,
u=sin(4x),
du=4cos(4x)dx)}}}{{{matrix(3,1,
dv=sin(x),
v=int(sin(x),dx),
v=-cos(x))}}}

{{{I}}}{{{""=""}}}{{{int(sin(x)*sin(4x)dx)}}}{{{""=""}}}{{{u*v-int(v*du)}}}{{{""=""}}}
{{{-sin(4x)cos(x)-int((-cos(x)^"")(4cos(4x)dx^""))}}}{{{""=""}}}
{{{-sin(4x)cos(x)+4int(cos(x)cos(4x)dx)}}}

So let's write and mark what we have so far, so we can come back to
it and substitute:
-----------------------------------------------------------
{{{I}}}{{{""=""}}}{{{-sin(4x)cos(x)+4int(cos(x)(cos(4x)dx))}}}
-----------------------------------------------------------
We now have find {{{int(cos(x)cos(4x)dx)}}}

But we have to be careful here, so we don't "undo" what we just did.
For this part, be sure to let u equal to the part that came from
what we let u equal before.

{{{matrix(2,1,
u=cos(4x),
du=-4sin(4x)dx)}}}{{{matrix(3,1,
dv=cos(x),
v=int(cos(x),dx),
v=sin(x))}}}

{{{int(cos(x)*cos(4x)dx)}}}{{{""=""}}}{{{u*v-int(v*du)}}}{{{""=""}}}
{{{cos(4x)sin(x)-int((sin(x)^"")(-4sin(4x)dx^""))}}}{{{""=""}}}
{{{cos(4x)sin(x)+4int(sin(x)sin(4x)dx)}}}

Let's write what we have here.
{{{int(cos(x)*cos(4x)dx)}}}{{{""=""}}}{{{cos(4x)sin(x)+4int(sin(x)sin(4x)dx)}}}

That integral on the right is just the original integral that we called I:

{{{int(cos(x)*cos(4x)dx)}}}{{{""=""}}}{{{cos(4x)sin(x)+4I}}}

So we go back to

{{{I}}}{{{""=""}}}{{{-sin(4x)cos(x)+4int(cos(x)(cos(4x)dx))}}}

and substitute  cos(4x)sin(x)+4I for the last integral:

{{{I}}}{{{""=""}}}{{{-sin(4x)cos(x)+4(cos(4x)^""sin(x)+4I)}}}

{{{I}}}{{{""=""}}}{{{-sin(4x)cos(x)+4cos(4x)^""sin(x)+16I}}}

Now we solve for I

{{{-15I}}}{{{""=""}}}{{{-sin(4x)cos(x)+4cos(4x)sin(x)}}}

{{{I}}}{{{""=""}}}{{{expr(1/15)sin(4x)cos(x)-expr(4/15)cos(4x)sin(x)}}}

But don't forget to put +C on the end.  
[It isn't necessary to put +C in the integration steps.]

{{{I}}}{{{""=""}}}{{{expr(1/15)sin(4x)cos(x)-expr(4/15)cos(4x)sin(x)+C}}} 

Edwin</pre>