Integral of $\sin|x|$

Question

$$\int\sin|x|~dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x)~dx+\int_0^\infty\sin x~dx$$ Left side of this sum is equals to right side, so we have just $$2\int_0^\infty\sin x~dx$$ Now, the integral of $\sin x$ seems to be incalculable.

Answer 1

Are you looking for the indefinite integral (anti derivative) or the definite integral from $-\infty$ to $\infty$? If it's the latter:

$$ \int_0^{\infty} \sin x \,dx = \lim_{t \rightarrow \infty} \left(-\cos x \right)\Big|_0^t = 1 - \lim_{t \rightarrow \infty} \,\cos t $$

The limit does not exist, so the integral diverges

For the anti derivative:

$$ \sin |x| = \left\{ \begin{array}{l l} -\sin x & \quad x < 0\\ \sin x & \quad x > 0 \end{array} \right. $$

So

$$ \int \sin|x| \,dx = \left\{ \begin{array}{l l} \cos x + C & \quad x < 0\\ \text{undefined} & \quad x = 0 \\ -\cos x + C & \quad x > 0 \end{array} \right. $$