Wednesday, January 22, 2014

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

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`



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