Question 1 :
$\int { { ({ x }^{ 2 }+5) }^{ 3 } } dx$
Question 3 :
If $I =\displaystyle \int {\dfrac{{dx}}{{{{\left( {2ax + {x^2}} \right)}^{\frac{3}{2}}}}}} $, then $I$ is equal to
Question 4 :
Assertion: If $\displaystyle \Delta (x)= \begin{vmatrix}f(x) &g(x) \\m_{1} &m_{2} \end{vmatrix}$ then<br><br>$\displaystyle \int \Delta (x)=\begin{vmatrix}<br><br>\int f(x)dx &\int g(x)dx \\m_{1} &m_{2}\end{vmatrix}$
Reason: $\displaystyle \int \lambda f(x)dx=\lambda\int f(x)dx$
Question 5 :
Let $f$ be a function which is continuous and differentiable for all real $x$. If $f\left( 2 \right) = -4$ and $f^{ \prime }\left( x \right) \ge 6$ for all $x\in \left[ 2,4 \right] $, then
Question 6 :
Evaluate: $\displaystyle \int { \dfrac { x\sqrt { x } .dx }{ \sqrt { 1-{ x }^{ 5 } }  } } $
Question 7 :
Solve: $\int \dfrac{2 \sin 2\ x - \cos \ x}{6 - \cos^2 \ x - 4 \sin \ x}d\theta$.
Question 9 :
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 integral $\displaystyle \int { \left( 1+2{ x }^{ 2 }+\frac { 1 }{ x } \right) } { e }^{ { x }^{ 2-\frac { 1 }{ x } } }dx$ is equal to
Question 16 :
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 17 :
The solution of $\displaystyle \frac { dy }{ dx } =\left( \frac { ax+b }{ cy+d } \right)$ represents aparabola if :-
Question 18 :
The integral $\displaystyle \int{\frac{\sec^2 x}{\left(\sec x + \tan x \right)^{9/2}}}$ dx equals (for some arbitrary constant $k$)<br>
Question 19 :
$\int \dfrac{cos 2x - cos 2 \theta}{cos x - cos \theta} dx$ is equal to
Question 20 :
$\displaystyle \int { \frac { \sqrt { x }  }{ \sqrt { { x }^{ 3 }+4 }  } dx } $ is equal to
Question 21 :
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 23 :
If $f\left( \cfrac { 3x-4 }{ 3x+4 } \right) =x+2$, then $\int { f(x) } dx$ is
Question 24 :
$\int { \cfrac { 1 }{ 8\sin ^{ 2 }{ x } +1 } } dx$ is equal to
Question 25 :
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