Common Integration Trick

Sine is an odd function. Sum of several odd functions is odd. Dividing it by an even function leaves the function odd.

Integral of an odd function over an interval symmetrical with respect to the origin is zero.

$\begin{aligned} I & = \int_{-\pi}^\pi \frac {\sum_{n=1}^9 \sin (nx)}{\ln (x^2+e)} dx \\ & = \sum_{n=1}^9 \color{#3D99F6} \int_{-\pi}^\pi \frac {\sin (nx)}{\ln (x^2+e)} dx & \small \color{#3D99F6} \text{Note that }\sin (nx) \text{ is odd and }\ln(x^2+e) \text{ is even, } \\ & = \sum_{n=1}^9 \color{#3D99F6} 0 & \small \color{#3D99F6} \text{therefore the integrand }\frac {\sin (nx)}{\ln (x^2+e)} \text{ is odd.} \\ & = \boxed{0} \end{aligned}$