Definite Integrals

Q.

Solve the integral $I={\int }_{\infty }^R\frac{GMm}{x^2}dx$

1. - GMm ln(R)

2. 

3. 

4. 

Q. A particle executes simple harmonic motion represented by displacement function as $x\left(t\right)=A\sin (\omega t+\varphi ).$ If the position and velocity of the particle at $t=0\text{s}$ are $2\text{cm}$ and $2\omega {\text{cm s}}^{-1}$ respectively, then its amplitude is $x\sqrt{2}\text{cm}$ where the value of $x$ is

Q. Evaluate : ${\int }_1^4(\frac{8}{\sqrt{t}}-12\sqrt{t^3})dt$

1. $-\frac{32}{5}$
2. $-\frac{608}{5}$
3. $-\frac{640}{5}$
4. $-\frac{664}{5}$

Q. Evaluate: ${\int }_2^5\text{(ln x)dx}$

1. $5ln5+2ln2-3$
2. $5ln5-2ln2-7$
3. $5ln5-2ln2+3$
4. $5ln5-2ln2-3$

Q. The area bounded by the curve $y=x^2+x+4$, the x-axis and the ordinates $x=1$ and $x=3$ is

Q. A spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is:

1. $\frac{9}{2}$
2. $\frac{9}{7}$
3. $\frac{7}{9}$
4. $\frac{2}{9}$

Q.

The area of the triangle formed by the line $\mathrm{x}\sin \mathrm{\alpha }+\mathrm{y}\cos \mathrm{\alpha }=\sin 2\mathrm{\alpha }$ and the coordinates axes is

1. $\sin 2\mathrm{\alpha }$

2. $\cos 2\mathrm{\alpha }$

3. $2\sin 2\mathrm{\alpha }$

4. $2\cos 2\mathrm{\alpha }$

Q. The area bounded by the curve $y=x^2+4$, the x-axis and the ordinates $x=1$ and $x=4$ is units.

Q.

Find the area enclosed by the curve $y=x^2$ and the x-axis for interval x = 1 to x = 3.

1. 9 sq. units

2. 27.2 sq. units

3. 10 sq. units

4. $\frac{26}{3}$ sq. units

Q. In the solar system, the Sun is in the focus of the ellipse for Sun-earth binding system. The binding energy for the system will be $($Given that radius of the earth's orbit around the Sun is $1.5\times {10}^{11}\text{m}$, mass of the earth is $6\times {10}^{24}\text{kg}$, mass of the Sun is $6\times {10}^{30}\text{kg}$, and $G=6.67\times {10}^{-11}{\text{Nm}}^2/$${\text{kg}}^2)$

1. $2.7\times {10}^{33}\text{J}$
   
2. $5.4\times {10}^{33}\text{J}$
3. $2.7\times {10}^{23}\text{J}$
4. $5.4\times {10}^{32}\text{J}$

Q. A container is filled with liquid upto height $H$ as shown in figure. There are $2$ holes in the container such that water coming out from them have same range. If the range of water coming out of the holes is half of maximum range, then the distance $x$ between these holes is

1. $\frac{\sqrt{5}H}{4}$
2. $\frac{\sqrt{3}H}{4}$
3. $\frac{2H}{3}$
4. $\frac{\sqrt{3}H}{2}$

Q. If $x=(6y+4)(3y^2+4y+3)$, then $\int xdy$ will be

1. $\frac{1}{3y^2+4y+3}+C$
2. $\frac{(3y^2+4y+3{)}^2}{2}+C$
3. $3y^2+4y+3+C$
4. $\frac{(6y+4)}{3y^2+4y+3}+C$

Q. The area bounded by the curve $y=x^2+4$, the x-axis and the ordinates $x=1$ and $x=4$ is units.

Q.

A piece of metal weight 46 gm in air, when it is immersed in the liquid of specific gravity 1.24 at 27ºC it weighs 30 gm. When the temperature of liquid is raised to 42ºC the metal piece weight 30.5 gm, specific gravity of the liquid at 42ºC is 1.20, then the linear expansion of the metal will be

1. 2.316 × 10^-5/ºC

2. 4.316 × 10^-5/ºC

3. None of these

4. 3.316 × 10^-5/ºC

Q.

Find an expression for $\cos 3x$ in terms of $\cos x$.

Q. ${\int }_0^1(e^x+x^2)dx$ is equal to

1. $e-\frac{2}{3}$
2. $e+\frac{2}{3}$
3. $e-1$
4. $e+1$

Q. This $Sl$ unit of inductance, the henry can be written as:

1. volt-second/ampere
2. weber/ampere
3. joule/$(ampere{)}^2$
4. ohm-second

Q.

If the displacement $x$ and velocity $v$ of a particle executing simple harmonic motion are related through the expression $4v^2=25-x^2$, then its time period is

1. $\pi$

2. $2\pi$

3. $4\pi$

4. $6\pi$

Q. If $a=(3t^2+2t+1)m/s^2$ is the expression according to which the acceleration of a particle moving along a straight line varies, then the expression for instantaneous velocity at any time $t$ will be (if the particle was initially at rest)

1. $t^3+2t+1$
2. $t^3+t+1$
3. $t^3+t^2+t$
4. $t^3+t^2+t+1$

Q. If $a=(3t^2+2t+1)m/s^2$ is the expression according to which the acceleration of a particle moving along a straight line varies, then find displacement of the particle 2 seconds after starting.

1. $26m$
2. $\frac{26}{3}m$
3. $\frac{30}{7}m$
4. $\frac{26}{7}m$

Q. If $a=(3t^2+2t+1)m/s^2$ is the expression according to which the acceleration of a particle moving along a straight line varies, then find displacement of the particle 2 seconds after starting.

1. $26m$
2. $\frac{26}{3}m$
3. $\frac{30}{7}m$
4. $\frac{26}{7}m$

Q. Given equation of the curve is $y=sinx$.

Using definite integral, find the area of the region between the given curve and the x-axis in the interval of $[0,\pi ]$.

1. $1$
2. $2$
3. $\frac{\pi }{2}$
4. $\pi$

Q. Evaluate:
i. ${\int }_0^{\frac{\pi }{2}}\text{(sin x + cos x)dx}$
ii. ${\int }_0^1(3x^2+4)dx$
iii. ${\int }_0^1(3e^{3x}+e^{-x})dx$

1. i. 0; ii. 5; iii. $\frac{e^4-1}{e}$
2. i. 0; ii. 5; iii. $e^3-1$
3. i. 2; ii. 5; iii. $e^3-1$
4. i. 2; ii. 5; iii. $\frac{e^4-1}{e}$

Q.

You are given a rod of length 'L'. The linear mass density varies as ${\lambda }^{\prime }$. $\lambda =a+bx$. Here a & b are constants and the mass of the rod increases as x increases. Find the mass of the rod.

1. 

2. 

3. 

4. None of these

Q. The value of ${\int }_1^5{x}^{2}dx$ is

1. $45$
2. $\frac{125}{3}$
3. $\frac{1}{3}$
4. $\frac{124}{3}$

Q. If $a=(3t^2+2t+1)m/s^2$ is the expression according to which the acceleration of a particle moving along a straight line varies, then the change in velocity after 3 seconds of its start is

1. $30m/s$
2. $39m/s$
3. $3m/s$
4. $20m/s$

Q. Can I differentiate any other function except $x^3$ to get $3x^2$.

1. No
2. Yes

Q. A particle executes simple harmonic motion between $x=-A$ and $x=+A$. The time taken for it to go from the mean position $O$ to $A/2$ is $T_1$ and to go from $A/2$ to $A$ is $T_2$. Then

1. $T_1<T_2$
2. $T_1>T_2$
3. $T_1=T_2$
4. $T_1=2T_2$

Q. Solve $y=\left|\cos x\right|,y=0,x=-\pi$ and $x=\pi$

Q. Evaluate :${\int }_0^1\frac{e^{5lnx}-e^{4lnx}}{e^{3lnx}-e^{2lnx}}dx$

1. $\frac{1}{3}$
2. 1
3. 0
4. $\frac{1}{2}$