How to find a definite integral over a symmetric interval without finding the antiderivative? How do I find the following without finding the anti derivative
$$
\int_{-\pi}^\pi \ln(x^2+1)e^{\sin \lvert x\rvert}\sin x dx 
$$ 

Interval $[-\pi,\pi]$ is symmetric about the origin and
$$\ln(x^2+1)e^{\sin \lvert x\rvert}$$
is even, but $\sin x$ is odd, thus the integrand function is odd. Then, the integral is $0$.
