Question

How do you integrate `∫3sin(ln(x))dx`?

Answer 1

Hello,

Answer. `3/2 x (sin(ln(x)) -cos(ln(x))) + c`, where `c in RR`.

Use complex numbers.

`int 3 sin(ln(x)) dx = 3 \mathfrak{Im} \int e^{i ln (x)} dx = 3 \mathfrak{Im}\int x^(i) dx`.

But `int x^(i) dx = x^(i+1)/(i+1) + c = x (e^(i ln(x))(1-i))/((1+i)(1-i)) + c = x/2 (e^(i ln(x))(1-i))`

Now, you extract the imaginary part of `e^(i ln(x))(1-i)` :

`e^(i ln(x))(1-i) = (cos(ln(x)) + i sin(ln(x)))(1-i)`

`e^(i ln(x))(1-i) = i(-cos(ln(x)) + sin(ln(x))) + \text{something real}`

Therefore, `mathfrak{Im}(e^(i ln(x))(1-i)) = -cos(ln(x)) + sin(ln(x))`.

Conclusion.

`int 3 sin(ln(x)) dx = 3 x/2(-cos(ln(x)) + sin(ln(x)))`.