If $f(a+b-x)=f(x)$,then $\int_a^b x f(x) d x=$ . . . . . . .

$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

If $\int_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x \, dx}{(1+e^{\sin x})(1+\sin ^4 x)} = \alpha \pi + \beta \log _e(3+2 \sqrt{2})$,where $\alpha, \beta$ are integers,then $\alpha^2+\beta^2$ equals.....................

If $f:R \to R$ and $g:R \to R$ are one-to-one,real-valued functions,then the value of the integral $\int_{-\pi}^{\pi} [f(x) + f(-x)][g(x) - g(-x)] \, dx$ is

Value of the definite integral $\int_{-1/2}^{1/2} (\sin^{-1}(3x - 4x^3) - \cos^{-1}(4x^3 - 3x)) dx$

$\int_{1/2}^{2} \frac{1}{x} \sin \left( x - \frac{1}{x} \right) dx = $