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

If $h(a) = h(b)$,the value of the integral $\int_a^b {[f(g(h(x)))]^{-1} f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \, dx} = $

$\int\limits_0^\pi (x \cdot \sin^2 x \cdot \cos x) dx =$

Assertion $(A)$: $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{(\sin x)^{\sqrt{2}} dx}{(\sin x)^{\sqrt{2}}+(\cos x)^{\sqrt{2}}} = \frac{\pi}{12}$
Reason $(R)$: $\int_{a}^{b} \frac{f(x) dx}{f(x)+f(a+b-x)} = \frac{b-a}{2}$

If $[t]$ denotes the greatest integer $\leq t$,then the value of $\frac{3(e-1)^2}{e} \int \limits_1^2 x^2 e^{[x]+[x^3]} dx$ is:

$\int_0^{\pi /2} \frac{d\theta}{1 + \tan \theta} = $