$\int_{0}^{\pi /2} \{ x - [\sin x] \} \,dx$ is equal to

Let $I = \int_{10}^{19} \frac{\sin x}{1+x^{6}} dx$. Then,

Consider the matrices $A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}$,$B = \begin{bmatrix} 20 \\ m \end{bmatrix}$,and $X = \begin{bmatrix} x \\ y \end{bmatrix}$. Let the set of all $m$,for which the system of equations $AX = B$ has a negative solution (i.e.,$x < 0$ and $y < 0$),be the interval $(a, b)$. Then $8 \int_a^b |A| dm$ is equal to.............

Let $f(x) + 2f\left(\frac{1}{x}\right) = x^2 + 5$ and $2g(x) - 3g\left(\frac{1}{x}\right) = x$ for $x > 0$. If $\alpha = \int_1^2 f(x) dx$ and $\beta = \int_1^2 g(x) dx$,then the value of $9\alpha + \beta$ is:

If $I_{1}=\int_{0}^{\pi / 2} x \sin x \, dx$ and $I_{2}=\int_{0}^{\pi / 2} x \cos x \, dx$,then which one of the following is true?

$A$ value of $\alpha$ such that $\int_{\alpha}^{\alpha+1} \frac{dx}{(x+\alpha)(x+\alpha+1)} = \log_{e}\left(\frac{9}{8}\right)$ is