Let $f:(0, \infty) \rightarrow \mathbb{R}$ and $F(x)=\int_0^x t f(t) d t$. If $F(x^2)=x^4+x^5$,then $\sum_{r=1}^{12} f(r^2)$ is equal to :

$A$ quadratic polynomial $P(x)$ satisfies the conditions $P(0) = 0$,$P(1) = 0$,and $\int_{0}^{1} P(x) dx = 1$. The leading coefficient of the quadratic polynomial is:

The velocity of a particle which starts from rest is given by the following table. The total distance travelled (in metre) by the particle in $10 \ s$, using the Trapezoidal rule, is:

$t \ (\text{in second})$	$0$	$2$	$4$	$6$	$8$	$10$
$v \ (\text{in m/s})$	$0$	$12$	$16$	$20$	$35$	$60$

If $f(x) = \frac{|\log x|}{x^2}$, then $\int_{1/e}^e f(x) dx =$

$\int_0^{\pi / 2} \sqrt{\sin \theta} \cos ^3 \theta d \theta$ is equal to

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x) = \begin{cases} x^2+3x+a, & x \leq 1 \\ bx+2, & x > 1 \end{cases}$ is differentiable on $\mathbb{R}$. Then,the value of $\int_{-2}^2 f(x) dx$ equals