Definite Integrals

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

1. No
2. Yes

Q.

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

1. - GMm ln(R)

2. 

3. 

4. 

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. 

2. 

3. 

4. None of these

Q.

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

1. None of these

2. 1 unit

3. 2 units

4. 0 unit

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.

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

1. 0

2. 

3. 

4. 

Q.

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

1. 

2. 

3. None of these

4. 

Q.

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

1. None of these

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

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

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

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

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