Standard integrals

Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}=\left(1+x+x^2\right)\left(1-y+y^2\right)$, $y(0)=\dfrac12$. Then $(2y(1)-1)$ is equal to

If

$$\alpha = \int_{1/2}^{2}\frac{\tan^{-1}x}{2x^2 - 3x + 2}\,dx,$$

then the value of $\sqrt7\tan\left(\dfrac{2\alpha\sqrt7}{\pi}\right)$ is _____.

(Here, the inverse trigonometric function $\tan^{-1}x$ assumes values in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.)

If $\displaystyle\int\frac1x\sqrt{\frac{1-x}{1+x}}\,dx=g(x)+c$, $g(1)=0$, then $g\left(\dfrac12\right)$ is equal to :

The integral $\displaystyle\int\frac{\left(1-\frac1{\sqrt3}\right)(\cos x-\sin x)}{\left(1+\frac2{\sqrt3}\sin2x\right)}dx$ is equal to

For $I(x)=\displaystyle\int\frac{\sec^2x-2022}{\sin^{2022}x}\,dx$, if $I\left(\dfrac\pi4\right)=2^{1011}$, then

The integral $\displaystyle\int\sqrt{1+2\cot x(\operatorname{cosec}x+\cot x)}\,dx$ $\left(0<x<\dfrac\pi2\right)$ is equal to :

(where C is a constant of integration)

If $\displaystyle\int_1^2\frac{dx}{(x^2-2x+4)^{3/2}}=\frac k{k+5}$, then k is equal to :

The integral $\displaystyle\int\left(1+x-\frac1x\right)e^{x+\frac1x}\,dx$ is equal to :

$\displaystyle\int\frac{\sin^8x-\cos^8x}{1-2\sin^2x\cos^2x}\,dx$ is equal to :

Let $F(x)$ be an indefinite integral of $\sin^2x$.

STATEMENT-1: The function $F(x)$ satisfies $F(x+\pi)=F(x)$ for all real $x$.

because

STATEMENT-2: $\sin^2(x+\pi)=\sin^2x$ for all real $x$.

Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.