Definite Integrals

Q.

Where should the tank be punctured so that water coming out has maximum horizontal range?

1. $y=\frac{h}{2}$

2. $y=\frac{h}{3}$

3. $y=\frac{h}{4}$

4. None of these

Q. Evaluate: ${\int }_0^{\frac{\pi }{2}}\left(\text{7 sin t - 2 cos t}\right)dt$

1. 5
2. 2
3. 7
4. \N

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

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

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

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

Q. Evaluate ${\int }_0^1\frac{1}{\sqrt{1+x}-\sqrt{x}}dx$

1. $\frac{4\sqrt{2}}{3}$
2. $2\sqrt{2}-2$
3. $\frac{4}{3}[\sqrt{2}-1]$
4. $\frac{2}{3}[2\sqrt{2}-1]$

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.

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

3. 27.2 sq. units

4. 10 sq. units

Q.

The velocity v and displacement x of a particle executing simple harmonic motion are related as

$v\frac{dv}{dx}=-{\omega }^{2}x$

At $x=0,v=v_0$. Find the velocity v when the displacement becomes x.

1. $v=\sqrt{v_0^2-{\omega }^2{x}^2}$

2. $\sqrt{v_0^2+{\omega }^2{x}^2}$

3. $v=\sqrt{v_0-{\omega }_0^{2}x}$

4. None of these

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. 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.

${\int }_{-\pi /2}^{\pi /2}cosxdx$

1. 1 unit

2. 2 units

3. 0 unit

4. None of these

Q.

${\int }_{-\pi /2}^{\pi /2}cosxdx$

1. 1 unit

2. 2 units

3. 0 unit

4. None of these

Q. Evaluate ${\int }_0^1\frac{1}{\sqrt{1+x}-\sqrt{x}}dx$

1. $\frac{4\sqrt{2}}{3}$
2. $2\sqrt{2}-2$
3. $\frac{4}{3}[\sqrt{2}-1]$
4. $\frac{2}{3}[2\sqrt{2}-1]$

Q.

${\int }_0^{10}se{c}^2(3x+6)dx$

1. $\frac{1}{3}tan\left({36}^o\right)$

2. $\frac{1}{3}tan\left(6^o\right)$

3. $\frac{1}{3}tan\left({30}^o\right)$

4. None of these

Q. Find ${\int }_{x=\infty }^{x=R}\frac{GMm}{x^2}dx$ $(G,M$ and $m$ are constants)

1. $\frac{GMm}{R}$
2. $-\frac{GMm}{R}$
3. $-\frac{GMm}{R^2}$
4. $-\frac{GMm}{R^2}$

Q. Evaluate: ${\int }_0^{\pi }(cosx+x^2)dx$

1. $\frac{{\pi }^3}{3}-1$
2. $\frac{{\pi }^3}{3}$
3. $\frac{{\pi }^3}{3}+1$
4. \N

Q. Solve the integral $I={\int }_0^{\pi }{\sin }^{2}xdx$

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

Q. Find ${\int }_{x=\infty }^{x=2R}\frac{GMm}{x^2}dx$ $(G,M$ and $m$ are constants)

1. $\frac{GMm}{R}$
2. $-\frac{GMm}{2R}$
3. $-\frac{GMm}{R}$
4. $-\frac{GMm}{R^2}$

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.

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

3. 27.2 sq. units

4. 10 sq. units

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

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

Q.

Solve the integral I = ${\int }_0^{\pi }si{n}^{2}xdx$

1. $\pi$

2. $\frac{\pi }{2}$

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

4. 0

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

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

Q. Evaluate: ${\int }_0^{\pi }(cosx+x^2)dx$

1. $\frac{{\pi }^3}{3}-1$
2. $\frac{{\pi }^3}{3}$
3. $\frac{{\pi }^3}{3}+1$
4. \N

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

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