If $[.]$ represents the greatest integer function,then evaluate the integral: $\int_{\frac{3 \pi}{4}}^\pi \left[ \sin x + \left[ \frac{4 x}{\pi} \right] \right] dx$.

If $\int_a^b {x^3} dx = 0$ and $\int_a^b {x^2} dx = \frac{2}{3}$,then the values of $a$ and $b$ are respectively:

Let $I = \int_{0}^{1} \frac{\sin x}{\sqrt{x}} \, dx$ and $J = \int_{0}^{1} \frac{\cos x}{\sqrt{x}} \, dx$. Then which one of the following is true?

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.............

If $\int_0^b \frac{dx}{1+x^2} = \int_b^{\infty} \frac{dx}{1+x^2}$, then $b$ is equal to

The value of the integral $\int \limits_1^2 \left(\frac{t^4+1}{t^6+1}\right) dt$ is $..........$.