Question 3 :
$\displaystyle \int { \frac { \sqrt { x }  }{ \sqrt { { x }^{ 3 }+4 }  } dx } $ is equal to
Question 4 :
A tank initially holds 10 lit. of fresh water. At t = 0, a brine solution containing $\displaystyle \frac{1}{2}$ kg of salt per lit. is poured into tank at a rate 1 lit/min while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt in a tank at a particular time
Question 5 :
The average value of pressure varying from 2 to 10 atm if the pressure $p $ and the volume $v$ are related by $ pv^{3/2} = 160 $ is
Question 6 :
Integrate <br/>$\displaystyle\int {\dfrac{{dx}}{{\left( {x + 1} \right)\sqrt {2{x^2} + 3x + 1} }}} $
Question 9 :
lf $\displaystyle \int f(x)\sin x\cos x\>dx=\frac{1}{2(b^{2}-a^{2})}\log(f(x))+c$, then $\displaystyle f(x)$ is equal to<br>
Question 10 :
A tank initially holds 10 lit. of fresh water. At t = 0, a brine solution containing $\displaystyle \frac{1}{2}$ kg of salt per lit. is poured into tank at a rate 1 lit/min while the well-stirred mixture leaves the tank at the same rate. Find the concentration of salt in the tank at any time $t$.
Question 11 :
The solution of $\displaystyle \frac { dy }{ dx } =\left( \frac { ax+b }{ cy+d } \right)$ represents aparabola if :-
Question 13 :
If $\displaystyle l^{r}(x)$ means $\log \log \log .........x,$ the $\log $ being repeated r times, then$\displaystyle \int \left [x l(x)l^{2}(x)l^{3}(x)......l^{r}(x) \right ]^{-1}dx$is equal to<br>
Question 14 :
$\displaystyle\int { \sin ^{ -1 }{ \sqrt { \dfrac { x }{ a+x } } } dx } $ is equal to
Question 15 :
If $I =\displaystyle \int {\dfrac{{dx}}{{{{\left( {2ax + {x^2}} \right)}^{\frac{3}{2}}}}}} $, then $I$ is equal to
Question 16 :
$\displaystyle \int \dfrac { \cot ^ { 2 } x } { \left( cosec ^ { 2 } x + cosec x \right) } d x =$
Question 17 :
$\int \dfrac{cos 2x - cos 2 \theta}{cos x - cos \theta} dx$ is equal to
Question 18 :
Solve: $\int \dfrac{2 \sin 2\ x - \cos \ x}{6 - \cos^2 \ x - 4 \sin \ x}d\theta$.
Question 19 :
The integral $\displaystyle \int{\frac{\sec^2 x}{\left(\sec x + \tan x \right)^{9/2}}}$ dx equals (for some arbitrary constant $k$)<br>