Enter '1' if following statement is true otherwise enter '0'.

$\int a^{x+{\tan }^{-1}x}\left[\frac{x^2+2}{x^2+1}\right]dx$$=\frac{a^{x+{\tan }^{-1}\left(x\right)}}{\ln \left(a\right)}+c$

Q. $\int \frac{dx}{x^4\sqrt{a^2+x^2}}=\frac{1}{a^4}[\frac{1}{x}\sqrt{a^2+x^2}-\frac{1}{3x^3}{(a^2+x^2)}^{3/2}]$. If this is true enter 1, else enter 0.

Q. Inverse circular functions,Principal values of ${sin}^{-1}x,{cos}^{-1}x,{tan}^{-1}x$.
${tan}^{-1}x+{tan}^{-1}y={tan}^{-1}\frac{x+y}{1-xy}$, $xy<1$
$\pi +{tan}^{-1}\frac{x+y}{1-xy}$, $xy>1$.
If $0<x<1$, then
$\sqrt{1+x^2}{[{\left\{xcos\right({cot}^{-1}x)+sin({cot}^{-1}x\left)\right\}}^2-1]}^{1/2}$ is equal to
(a) $\frac{x}{\sqrt{1+x^2}}$
(b) $x$
(c) $x\sqrt{1+x^2}$
(d) $\sqrt{1+x^2}$

Q.

equals

A. x tan⁻¹ (x + 1) + C

B. tan^{− 1} (x + 1) + C

C. (x + 1) tan⁻¹ x + C

D. tan⁻¹ x + C

Q. If $x=\frac{-b+\sqrt{b^2-4ac}}{2a}$; 'c' as the subject of the formula is $c=-a{x}^2-bx$ ( Enter 1 if true or 0 otherwise)

Q. Inverse circular functions,Principal values of ${sin}^{-1}x,{cos}^{-1}x,{tan}^{-1}x$.
${tan}^{-1}x+{tan}^{-1}y={tan}^{-1}\frac{x+y}{1-xy}$, $xy<1$
$\pi +{tan}^{-1}\frac{x+y}{1-xy}$, $xy>1$.
(a) $3{sin}^{-1}\frac{2x}{1+x^2}-4{cos}^{-1}\frac{1-x^2}{1+x^2}+2{tan}^{-1}\frac{2x}{1-x^2}=\frac{\pi }{3}$
(b) $sin\left[\right(1/5\left){cos}^{-1}x\right]=1$
(c) ${tan}^{-1}\sqrt{x^2+x}+{sin}^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$