If $\int_{-a}^a f(x) dx = \int_0^a f(x) dx + \int_0^a g(x) dx$,then $g(x) =$

The value of the integral $\int_{-1}^{1} \frac{d}{dx} \left( \tan^{-1} \frac{1}{x} \right) dx$ is

The value of the definite integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{\left(1+e^{x \cos x}\right)\left(\sin ^{4} x+\cos ^{4} x\right)}$ is equal to:

$\int_0^{\infty} (x^{12} + x^{-12}) \frac{\log x}{x} dx =$

$\int_{-\pi / 2}^{\pi / 2}(2 \sin |x|+\cos |x|) d x=$

Evaluate the definite integral: $\int_{\frac{1}{2}}^{2} \frac{x^2 \ln x}{(1+x^2)^3} dx$