Q 21 :

$\text{Let }f\left(x\right)=\int \frac{2x}{(x^2+1)(x^2+3)}dx.$ $\text{If }f\left(3\right)=\frac{1}{2}({\log }_{e}5-{\log }_{e}6),\text{ then }f\left(4\right)\text{ is equal to}$             [2023]

• ${\log }_{e}17-{\log }_{e}18$

   

• ${\log }_{e}19-{\log }_{e}20$

   

• $\frac{1}{2}({\log }_{e}19-{\log }_{e}17)$

   

• $\frac{1}{2}({\log }_{e}17-{\log }_{e}19)$

   

Q 22 :

Let $I\left(x\right)=\int \sqrt{\frac{x+7}{x}}dx$ and $I\left(9\right)=12+7{\log }_{e}7$. If $I\left(1\right)=\alpha +7{\log }_e(1+2\sqrt{2})$ then ${\alpha }^4$ is equal to _______ .           [2023]

Q 23 :

Let $f\left(x\right)=\int \frac{dx}{(3+4x^2)\sqrt{4-3x^2}},$$\left|x\right|$$<\frac{2}{\sqrt{3}}$. If $f\left(0\right)=0$ and $f\left(1\right)=\frac{1}{\alpha \beta }{\tan }^{-1}\left(\frac{\alpha }{\beta }\right)$, $\alpha ,\beta >0$, then ${\alpha }^2+{\beta }^2$ is equal to _______ .        [2023]

Q 24 :

If $\int \sqrt{\sec 2x-1} dx=\alpha {\log }_e$$|\cos 2x+\beta +\sqrt{\cos 2x(1+\cos \frac{1}{\beta }x)}|$$+$constant, then $\beta -\alpha$ is equal to _________ .              [2023]

Q 25 :

Let $f\left(x\right)$$=\int \frac{(2-x^2).e^x}{\left(\sqrt{1+x}\right){(1-x)}^{3/2}}dx.$ If $f\left(0\right)=0$, then $f\left(\frac{1}{2}\right)$ is equal to:             [2026]

• $\sqrt{3e}+1$

   

• $\sqrt{2e}-1$

   

• $\sqrt{2e}+1$

   

• $\sqrt{3e}-1$

   

Q 26 :

Let $f\left(t\right)$$=\int \left(\frac{1-\sin \left({\log }_{e}t\right)}{1-\cos \left({\log }_{e}t\right)}\right)dt,t>1.$

If $f\left(e^{\pi /2}\right)=-e^{\pi /2}\text{ and }f\left(e^{\pi /4}\right)=\alpha e^{\pi /4},$ then $\alpha$ equals           [2026]

• $-1-\sqrt{2}$

   

• $1+\sqrt{2}$

   

• $-1-2\sqrt{2}$

   

• $-1+\sqrt{2}$

   

Q 27 :

Let $f\left(x\right)=\int \frac{dx}{x^{\left(\frac{2}{3}\right)}+2x^{\left(\frac{1}{2}\right)}}$ be such that $f\left(0\right)=-26+24 {\log }_e\left(2\right).$ If $f\left(1\right)=a+b{\log }_e\left(3\right),$ where $a,b\in \mathrm{\mathbb{Z}}$, then a+b is equal to: [2026]

• -11

   

• -26

   

• -18

   

• -5

   

Q 28 :

If $\int {\left(\sin x\right)}^{\frac{{\displaystyle -11}}{{\displaystyle 2}}}{\left(\cos x\right)}^{\frac{{\displaystyle -5}}{{\displaystyle 2}}}dx=-\frac{p_1}{q_1}{\left(\cot x\right)}^{\frac{{\displaystyle 9}}{{\displaystyle 2}}}-$$\frac{p_2}{q_2}{\left(\cot x\right)}^{\frac{5}{2}}-\frac{p_3}{q_3}{\left(\cot x\right)}^{\frac{1}{2}}+$$\frac{p_4}{q_4}{\left(\cot x\right)}^{\frac{-3}{2}}+C$

where $p_i$ and $q_i$ are positive integers with $\gcd (p_i,q_i)=1$ for $i=1,2,3,4$, and C is the constant of integration, then $\frac{15p_1{p}_2{p}_3{p}_4}{q_1{q}_2{q}_3{q}_4}$ is equal to ______     [2026]

Q 29 :

Let f be a differentiable function satisfying $f\left(x\right)=1-2x+{\int }_0^x{e}^{(x-t)} f\left(t\right) dt, x\in \mathrm{ℝ}$ and let $g\left(x\right)={\int }_0^x(f{\left(t\right)+2)}^{15 }{(t-4)}^6{(t+2)}^{17} dt, x\in \mathrm{ℝ}.$ If p and q are respectively the points of local minima and local maxima of g, then the value of |p+q| is equal to ________   [2026]

Q 30 :

Let $f\left(x\right)=\int \frac{7x^{10}+9x^8}{{(1+x^2+2x^9)}^2}dx,x>0,$ $\underset{x\to 0}{\lim }f\left(x\right)=0$ and $f\left(1\right)=\frac{1}{4}$.

If $A=\left[\begin{array}{ccc}0& 0& 1\\ \frac{{\displaystyle 1}}{{\displaystyle 4}}& f\text{'}\left(1\right)& 1\\ {\alpha }^2& 4& 1\end{array}\right]$ and $B=\mathit{adj}(\mathit{adj}A)$ be such that $\left|B\right|$$=81$, then ${\alpha }^2$ is equal to               [2026]

• 1

   

• 4

   

• 2

   

• 3