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Maths-

General

Easy

Question

If $I equals integral x square root of fraction numerator x to the power of 2 end exponent plus 1 over denominator x to the power of 2 end exponent minus 1 end fraction end root d x comma text end text$ then I equals

$\frac{1}{2} \sqrt{x^{4} - 1} + \frac{1}{2} \sqrt{x^{4} + 1} + C$
$\frac{1}{2} \sqrt{x^{4} - 1} + \frac{1}{2} l o g (x^{2} + \sqrt{x^{4} - 1}) + C$
$\sqrt{x^{4} - 1} + {s i n}^{- 1} (x^{2}) + C$
$\sqrt{x^{4} - 1} + 2 {s i n}^{- 1} (x^{2}) + C$

The correct answer is: $fraction numerator 1 over denominator 2 end fraction square root of x to the power of 4 end exponent minus 1 end root plus fraction numerator 1 over denominator 2 end fraction l o g invisible function application open parentheses x to the power of 2 end exponent plus square root of x to the power of 4 end exponent minus 1 end root close parentheses plus C$

Related Questions to study

$stretchy integral subscript 0 end subscript superscript pi divided by 2 end superscript fraction numerator s i n to the power of 2 end exponent invisible function application 9 x over denominator sin invisible function application x end fraction d x equals$

$stretchy integral subscript 0 end subscript superscript pi divided by 2 end superscript fraction numerator s i n to the power of 2 end exponent invisible function application 9 x over denominator sin invisible function application x end fraction d x equals$

A square ABCD is inscribed in a circle of radius 4. A point P moves inside the circle such that $d left parenthesis P comma A B right parenthesis less or equal than$ min(d(P,BC),d(P,DA)) where d(P,AB) is the distance of a point P from line AB. The area of region covered by the moving point P is (sq units)

A square ABCD is inscribed in a circle of radius 4. A point P moves inside the circle such that $d left parenthesis P comma A B right parenthesis less or equal than$ min(d(P,BC),d(P,DA)) where d(P,AB) is the distance of a point P from line AB. The area of region covered by the moving point P is (sq units)

If $text end text I subscript n end subscript equals stretchy integral subscript 0 end subscript superscript infinity end superscript e to the power of negative x end exponent left parenthesis s i n invisible function application x right parenthesis to the power of n end exponent d x comma n greater than 1$ then the value of $fraction numerator 101 I subscript 10 end subscript over denominator I subscript 8 end subscript end fraction$ is equal to ____

If $text end text I subscript n end subscript equals stretchy integral subscript 0 end subscript superscript infinity end superscript e to the power of negative x end exponent left parenthesis s i n invisible function application x right parenthesis to the power of n end exponent d x comma n greater than 1$ then the value of $fraction numerator 101 I subscript 10 end subscript over denominator I subscript 8 end subscript end fraction$ is equal to ____

parallel

2k-mol of an ideal diatomic gas is enclosed in a vertical cylinder fitted with a piston and spring as shown in the figure. Initially the spring is compressed by 5 cm and then the electric heater starts supplying energy to the gas at constant rate of 100 J/S and due to conduction through walls of cylinder and radiation, 20 J/S has been lost to surroundings ( Take K=1000 N/m, $g equals 10 blank m divided by s to the power of 2 end exponent$ atomospheric pressure, $P subscript o end subscript equals 10 to the power of 5 end exponent blank N divided by m to the power of 2 end exponent$ , cross section area of piston $A subscript o end subscript equals 50 blank c m to the power of 2 end exponent$ , mass of piston m=1 kg R=8.3kj mol k) Answer the given question based on this information.

The initial pressure of the gas is

2k-mol of an ideal diatomic gas is enclosed in a vertical cylinder fitted with a piston and spring as shown in the figure. Initially the spring is compressed by 5 cm and then the electric heater starts supplying energy to the gas at constant rate of 100 J/S and due to conduction through walls of cylinder and radiation, 20 J/S has been lost to surroundings ( Take K=1000 N/m, $g equals 10 blank m divided by s to the power of 2 end exponent$ atomospheric pressure, $P subscript o end subscript equals 10 to the power of 5 end exponent blank N divided by m to the power of 2 end exponent$ , cross section area of piston $A subscript o end subscript equals 50 blank c m to the power of 2 end exponent$ , mass of piston m=1 kg R=8.3kj mol k) Answer the given question based on this information.

The initial pressure of the gas is

Physics-General

A system is taken from state ' i ' to the state 'f' along the path 'iaf', it is found that $capital delta Q equals 50 capital delta W$ =20 cal. Along the path 'ibf'. $capital delta Q equals 36$ cal.

If $capital delta W equals negative 13 c a l$ for curved path 'fi', what is the ' $capital delta Q to the power of ´ end exponent$ for this path ?

A system is taken from state ' i ' to the state 'f' along the path 'iaf', it is found that $capital delta Q equals 50 capital delta W$ =20 cal. Along the path 'ibf'. $capital delta Q equals 36$ cal.

If $capital delta W equals negative 13 c a l$ for curved path 'fi', what is the ' $capital delta Q to the power of ´ end exponent$ for this path ?

Physics-General

Find the efficiency of the thermodynamic cycle shown in figure for an ideal diatomic gas.

Find the efficiency of the thermodynamic cycle shown in figure for an ideal diatomic gas.

Physics-General

parallel

The temperature - entropy diagram of a reversible engine cycle is given in figure. What is its efficiency

The temperature - entropy diagram of a reversible engine cycle is given in figure. What is its efficiency

Physics-General

One mole of a monoatomic ideal gas undergoes the process A→B in the given P-V diagram. What is the specific heat for this process ?

One mole of a monoatomic ideal gas undergoes the process A→B in the given P-V diagram. What is the specific heat for this process ?

Physics-General

An ideal monoatomic gas is taken round the cycle ABCDA as shown in the P-V diagram. Compute the work done in this process.

An ideal monoatomic gas is taken round the cycle ABCDA as shown in the P-V diagram. Compute the work done in this process.

Physics-General

parallel

A block of ice at $negative 10 to the power of ring operator end exponent C$ is slowly heated and converted to steam at $100 to the power of ring operator end exponent C$ . Which of the following curves represents the phenomena qualitatively ?

A block of ice at $negative 10 to the power of ring operator end exponent C$ is slowly heated and converted to steam at $100 to the power of ring operator end exponent C$ . Which of the following curves represents the phenomena qualitatively ?

Physics-General

An enclosed ideal gas is taken through a cycle as shown in the figure. Then

An enclosed ideal gas is taken through a cycle as shown in the figure. Then

Physics-General

A student records $capital delta Q comma capital delta U & capital delta W$ for a thermodynamic cycle $A rightwards arrow B rightwards arrow C rightwards arrow A.$ Certain entries are missing. Find correct entry in following options.

A student records $capital delta Q comma capital delta U & capital delta W$ for a thermodynamic cycle $A rightwards arrow B rightwards arrow C rightwards arrow A.$ Certain entries are missing. Find correct entry in following options.

Physics-General

parallel

A resistance coil connected to an external battery is placed inside an adiabatic cylinder fitted with a frictionless piston and containing an ideal gas. A current i flows through the coil which has a resistance R. At what speed must the piston move upward in order that the temperature of the gas remains unchanged? Neglect atmospheric pressure.

A resistance coil connected to an external battery is placed inside an adiabatic cylinder fitted with a frictionless piston and containing an ideal gas. A current i flows through the coil which has a resistance R. At what speed must the piston move upward in order that the temperature of the gas remains unchanged? Neglect atmospheric pressure.

Physics-General

Consider the thermodynamics cycle shown on PV diagram The process $A rightwards arrow B$ is isobaric, $B rightwards arrow C$ is isochoric. The following internal energy and heat are given: $d U subscript A rightwards arrow B end subscript equals plus 400 text end text k J$ and $Q subscript B rightwards arrow C end subscript equals negative 500 text end text k J$ . The heat flow in the process $Q subscript C rightwards arrow A end subscript$ is :

Consider the thermodynamics cycle shown on PV diagram The process $A rightwards arrow B$ is isobaric, $B rightwards arrow C$ is isochoric. The following internal energy and heat are given: $d U subscript A rightwards arrow B end subscript equals plus 400 text end text k J$ and $Q subscript B rightwards arrow C end subscript equals negative 500 text end text k J$ . The heat flow in the process $Q subscript C rightwards arrow A end subscript$ is :

Physics-General

A closed container is fully insulated from outside. One half of it is filled with an ideal gas X separated by a plate P from the other half Y which contains a vacuum as shown in figure. When P is removed, X moves into Y. Which of the following statements is correct?

A closed container is fully insulated from outside. One half of it is filled with an ideal gas X separated by a plate P from the other half Y which contains a vacuum as shown in figure. When P is removed, X moves into Y. Which of the following statements is correct?

Physics-General

parallel

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