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 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:

If $(a, b)$ is the orthocentre of the triangle whose vertices are $(1, 2), (2, 3)$ and $(3, 1)$,and $I_1 = \int_{a}^{b} x \sin(4x - x^2) dx$,$I_2 = \int_{a}^{b} \sin(4x - x^2) dx$,then $36 \frac{I_1}{I_2}$ is equal to:

Let $\int\limits_0^1 {{{\tan }^{ - 1}}\left( {\frac{{\tan x}}{2}} \right)} dx = \alpha $. Then $\int\limits_0^1 {{{\tan }^{ - 1}}\left( {\frac{{\tan x - 2\cot x}}{3}} \right)} dx$ is equal to:

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(x^2 + \log \left(\frac{\pi-x}{\pi+x}\right) \cdot \cos x\right) dx =$

$\int_0^2 x^2(2-x)^5 d x=$