Indefinite Integration

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

The integral $\displaystyle\int_0^{1/2}\frac{\ln(1+2x)}{1+4x^2}\,dx$, equals :

The integral $\displaystyle\int x\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)dx$ $(x>0)$ is equal to :

If for $n\ge1$, $P_n=\displaystyle\int_1^e(\log x)^n\,dx$, then $P_{10}-90P_8$ is equal to :

The integral $\displaystyle\int\frac{\sin^2x\cos^2x}{(\sin^3x+\cos^3x)^2}dx$ is equal to :

If $m$ is a non - zero number and
$$\int\frac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3}dx=f(x)+c,$$
then $f(x)$ is :

The integral $\displaystyle\int\frac{\sec^2x}{(\sec x+\tan x)^{9/2}}\,dx$ equals (for some arbitrary constant $K$)

The value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2}\left(x^2+\ln\frac{\pi+x}{\pi-x}\right)\cos x\,dx$ is

The value of
$$\int_{\sqrt{\ln2}}^{\sqrt{\ln3}}\frac{x\sin x^2}{\sin x^2+\sin\left(\ln6-x^2\right)}\,dx$$
is

The value(s) of $\displaystyle\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx$ is (are)

The area of the region bounded by the curve $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$, is

The area of the region between the curves $y=\sqrt{\dfrac{1+\sin x}{\cos x}}$ and $y=\sqrt{\dfrac{1-\sin x}{\cos x}}$ bounded by the lines $x=0$ and $x=\frac\pi4$ is

Let
$$I=\int\frac{e^x}{e^{4x}+e^{2x}+1}dx,\qquad J=\int\frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx.$$
Then, for an arbitrary constant $C$, the value of $J-I$ equals

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.

Let $f(x)=\dfrac{x}{\left(1+x^n\right)^{1/n}}$ for $n\ge2$ and $g(x)=\underbrace{(f\circ f\circ\cdots\circ f)}_{f\text{ occurs }n\text{ times}}(x)$. Then $\displaystyle\int x^{n-2}g(x)\,dx$ equals