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Maths-

General

Easy

Question

$f left parenthesis x right parenthesis equals stretchy integral subscript 1 end subscript superscript x end superscript fraction numerator tan to the power of negative 1 end exponent invisible function application t over denominator t end fraction d t semicolon x greater than 0$ then the value of $f open parentheses e to the power of 2 end exponent close parentheses minus f open parentheses fraction numerator 1 over denominator e to the power of 2 end exponent end fraction close parentheses text is end text$

0
$\frac{π}{2}$
p
2p

The correct answer is: 2p

$f left parenthesis x right parenthesis equals stretchy integral subscript 1 end subscript superscript x end superscript fraction numerator Tan to the power of negative 1 end exponent invisible function application t over denominator t end fraction d t$
$table row cell f open parentheses fraction numerator 1 over denominator x end fraction close parentheses equals stretchy integral subscript 1 end subscript superscript 1 divided by x end superscript fraction numerator T a n to the power of negative 1 end exponent t over denominator t end fraction d t end cell row cell rightwards double arrow f open parentheses fraction numerator 1 over denominator x end fraction close parentheses equals negative stretchy integral subscript 1 end subscript superscript x end superscript fraction numerator cot to the power of negative 1 end exponent invisible function application t over denominator t end fraction d t rightwards double arrow f left parenthesis x right parenthesis minus f open parentheses fraction numerator 1 over denominator x end fraction close parentheses equals fraction numerator pi over denominator 2 end fraction stretchy integral subscript 1 end subscript superscript x end superscript fraction numerator d t over denominator t end fraction equals fraction numerator pi over denominator 2 end fraction l o g invisible function application x rightwards double arrow f open parentheses e to the power of 2 end exponent close parentheses minus f open parentheses fraction numerator 1 over denominator e to the power of 2 end exponent end fraction close parentheses equals 2 pi end cell end table$

Related Questions to study

$integral left parenthesis square root of c o t invisible function application x end root plus square root of t a n invisible function application x end root right parenthesis d x equals f left parenthesis x right parenthesis plus c text thenf end text left parenthesis x right parenthesis equals$

$integral left parenthesis square root of c o t invisible function application x end root plus square root of t a n invisible function application x end root right parenthesis d x equals f left parenthesis x right parenthesis plus c text thenf end text left parenthesis x right parenthesis equals$

If $text end text g left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis vertical line s i n invisible function application t vertical line plus vertical line c o s invisible function application t vertical line right parenthesis d t text then end text g open parentheses x plus fraction numerator n pi over denominator 2 end fraction close parentheses$ is equal to where $n element of N$

If $text end text g left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis vertical line s i n invisible function application t vertical line plus vertical line c o s invisible function application t vertical line right parenthesis d t text then end text g open parentheses x plus fraction numerator n pi over denominator 2 end fraction close parentheses$ is equal to where $n element of N$

If the primitive of $text end text s i n to the power of negative 3 divided by 2 end exponent invisible function application x s i n to the power of negative 1 divided by 2 end exponent invisible function application left parenthesis x plus theta right parenthesis text end text text is end text text end text minus 2 c o s e c invisible function application theta square root of f left parenthesis x right parenthesis end root plus C text end text$ then

If the primitive of $text end text s i n to the power of negative 3 divided by 2 end exponent invisible function application x s i n to the power of negative 1 divided by 2 end exponent invisible function application left parenthesis x plus theta right parenthesis text end text text is end text text end text minus 2 c o s e c invisible function application theta square root of f left parenthesis x right parenthesis end root plus C text end text$ then

parallel

$integral fraction numerator square root of s i n to the power of 3 end exponent invisible function application 2 x end root over denominator sin to the power of 5 end exponent invisible function application x end fraction d x text is end text$

$integral fraction numerator square root of s i n to the power of 3 end exponent invisible function application 2 x end root over denominator sin to the power of 5 end exponent invisible function application x end fraction d x text is end text$

Assertion : Two similar trains are moving along the equatorial line with the same speed but in opposite direction. They will exert equal pressure on the rails.
Reason : In uniform circular motion of magnitude of acceleration remains constant but the direction continuously changed.

Assertion : Two similar trains are moving along the equatorial line with the same speed but in opposite direction. They will exert equal pressure on the rails.
Reason : In uniform circular motion of magnitude of acceleration remains constant but the direction continuously changed.

Physics-General

Assertion : As the frictional force increases, the safe velocity limit for taking a turn on an unbanked road also increases.
Reason : Banking of roads will increase the value of limiting velocity.

Assertion : As the frictional force increases, the safe velocity limit for taking a turn on an unbanked road also increases.
Reason : Banking of roads will increase the value of limiting velocity.

Physics-General

parallel

A ball is rolled off the edge of horizontal table at a speed of 4m/sec. It hits the ground after 0.4 second. Which statement given below is true :

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Physics-General

A string of length L is fixed at one end and carries a mass M at the other end. The string makes 2/p revolutions per second around the vertical axis through the fixed end as shown in the figure, then tension in the string is :

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physics-General

A car moves on a circular road. It describes equal angles about the centre in equal intervals of time. Which of the following statement about the velocity of the car is true :

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Physics-General

parallel

A point mass m is suspended from a light thread of length l, fixed at O, is whirled in a horizontal circle at constant speed as shown. From your point of view, stationary with respect to the mass, the forces on the mass are :

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physics-General

Figure shows four paths for a kicked football. Ignoring the effects of air on the flight, rank the paths according to initial horizontal velocity component, highest first :

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physics-General

A stone is just released from the window of a train moving along a horizontal straight track. The stone will hit the ground following :

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parallel

Two racing cars of masses m₁ and m₂ are moving in circles of radii r₁ and r₂ respectively. Their speed are such that each makes a complete circle in the same duration of time t. The ratio of the angular speed of the first to the second car is :

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Physics-General

A ball is projected with kinetic energy E at an angle of 45 $degree$ to the horizontal. At the highest point during its flight, its kinetic energy will be :

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Physics-General

A fair die is tossed three times. Given that the sum of the first two tosses equals the third. The probability that at least one ‘2’ has turned up, is equal to

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parallel

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