If $I$ is the greatest of $I_1=\int_0^1 e^{-x} \cos ^2 x \, dx, I_2=\int_0^1 e^{-x^2} \cos ^2 x \, dx, I_3=\int_0^1 e^{-x^2} \, dx, I_4=\int_0^1 e^{-x^2 / 2} \, dx$, then

Let $f : [1/2, 1] \rightarrow \mathbb{R}$ be a positive,non-constant and differentiable function such that $f^{\prime}(x) < 2f(x)$ and $f(1/2) = 1$. Then the value of $\int_{1/2}^1 f(x) dx$ lies in the interval

Evaluate $\int_{-1}^{\frac{3}{2}}|x \sin (\pi x)| d x$

$\int_{1}^{x} \frac{\log(x^2)}{x} \, dx = $

Number of ordered pairs $(a, b)$ satisfying simultaneously the system of equations $\int_{a}^{b} x^{3} dx = 0$ and $\int_{a}^{b} x^{2} dx = \frac{2}{3}$ is

$\int_0^a \frac{x-a}{x+a} dx =$