If $\int_0^3 (3x^2 - 4x + 2) dx = k$, then an integer root of $3x^2 - 4x + 2 = \frac{3k}{5}$ is

$\int_0^{\pi /2} e^x \sin x \, dx = $

$\mathop \smallint \limits_0^\pi \sqrt {1 + 4{{\sin }^2}\frac{x}{2} - 4\sin \frac{x}{2}} \;dx = $

Evaluate the definite integral $\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}$.

$A$ 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 value of $\int_0^1 \frac{x^4 + 1}{x^2 + 1} \, dx$ is