`int sin(ln(x)) dx` First make a substitution and then use integration by parts to evaluate the integral

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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`

 

 

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