We need to make a substitution then use integration by parts.
Let us make the substitution:
`ln(x) = t,` so:
`x = e^t`
therefore `dx = e^t dt`
so our equation can be changed. `int sin(ln(x))dx = int e^t(sin(t)) dt`
Now use integration by parts.
Let `u = sin(t) and dv = e^tdt`
`du = cos(t) and v = e^t`
`int e^t sin(t)dt = e^tsin(t) - int e^tcos(t) dt`
We will call that equation 1.
...
We need to make a substitution then use integration by parts.
Let us make the substitution:
`ln(x) = t,` so:
`x = e^t`
therefore `dx = e^t dt`
so our equation can be changed. `int sin(ln(x))dx = int e^t(sin(t)) dt`
Now use integration by parts.
Let `u = sin(t) and dv = e^tdt`
`du = cos(t) and v = e^t`
`int e^t sin(t)dt = e^tsin(t) - int e^tcos(t) dt`
We will call that equation 1.
Now we need to evaluate that second integral with integration by parts again.
`int e^t cos(t) dt = e^t cos(t) - int e^t (-sin(t))dt`
`int e^t cos(t) dt = e^t cos(t) + int e^t sin(t)dt`
Now let us plug this result for int e^t cos(t) dt back into equation 1.
`int e^t sin(t)dt = e^tsin(t) - (e^t cos(t) + int e^t sin(t) dt)`
add `int e^t sin(t) dt ` to both sides:
`2 int e^t sin(t) dt = e^t sin(t) - e^t cos(t)`
sub back in our original `t = ln(x)` or `e^t = x.`
`2 int sin(ln(x)) dx = xsin(ln(x)) - xcos(ln(x))`
divide both sides by 2 and add the constant of integration. And were done!!!!
`int sin(ln(x)) dx = (xsin(ln(x)) - xcos(ln(x)))/2 + c`
No comments:
Post a Comment