Area bounded by curve $x y equals c comma x$ ‐axis between $x equals 1$ and $x equals 4$ is

The integral $not stretchy integral subscript 2 end subscript superscript 4 end superscript fraction numerator blank l o g blank x to the power of 2 end exponent over denominator I o g x to the power of 2 end exponent plus blank l o g blank left parenthesis 36 minus 12 x plus x to the power of 2 end exponent right parenthesis end fraction d x$ is equal to

The solution for $x$ of the equation $not stretchy integral subscript square root of 2 end subscript superscript x end superscript fraction numerator d r over denominator r square root of t to the power of 2 end exponent minus 1 end root end fraction equals fraction numerator pi over denominator 2 end fraction$ is

If the differential equation representing the family of all circles touching $x$ ‐axis at the origin is $left parenthesis x to the power of 2 end exponent minus y to the power of 2 end exponent right parenthesis fraction numerator d y over denominator d x end fraction equals g left parenthesis x right parenthesis y comma t h e n g left parenthesis x right parenthesis$ equals

Which one of the following is a differential equation of the family of curves $y equals A e to the power of 2 x end exponent plus B e to the power of negative 2 x end exponent$

A function $y equals f left parenthesis x right parenthesis$ has a second‐order derivatives $f to the power of ´ ´ end exponent left parenthesis x right parenthesis equals 6 left parenthesis x minus$ 1) If its graph passes through the point $left parenthesis 2 comma blank 1 right parenthesis$ and at that point the tangent to the grraph $i s y equals 3 x minus 5$ , then the function is

For such questions, the important part is integration. We should know the methods to integrate the derivate. We should know the properties of a tangent.

A function $y equals f left parenthesis x right parenthesis$ has a second‐order derivatives $f to the power of ´ ´ end exponent left parenthesis x right parenthesis equals 6 left parenthesis x minus$ 1) If its graph passes through the point $left parenthesis 2 comma blank 1 right parenthesis$ and at that point the tangent to the grraph $i s y equals 3 x minus 5$ , then the function is

For such questions, the important part is integration. We should know the methods to integrate the derivate. We should know the properties of a tangent.

The differential equation $y fraction numerator d y over denominator d x end fraction plus x equals o$ ( $0$ is any constant) represents

For such questions, we know different methods to solve differential equations. We should also know the formulas of different shapes.

The differential equation $y fraction numerator d y over denominator d x end fraction plus x equals o$ ( $0$ is any constant) represents

For such questions, we know different methods to solve differential equations. We should also know the formulas of different shapes.

An integrating factor of the differential equation $x fraction numerator d y over denominator d x end fraction plus y to the power of 1 end exponent o g x equals x e to the power of x end exponent x to the power of negative fraction numerator 1 over denominator 2 end fraction 1 o g x end exponent comma$ $left parenthesis x greater than 0 right parenthesis$ is

The solution of the differential equation $fraction numerator d y over denominator d x end fraction plus fraction numerator 3 x to the power of 2 end exponent over denominator 1 plus x to the power of 3 end exponent end fraction y equals fraction numerator s i n to the power of 2 end exponent x over denominator 1 plus x to the power of 3 end exponent end fraction$ is

For such questions, we should know different methods to solve differential equation.

The solution of the differential equation $fraction numerator d y over denominator d x end fraction plus fraction numerator 3 x to the power of 2 end exponent over denominator 1 plus x to the power of 3 end exponent end fraction y equals fraction numerator s i n to the power of 2 end exponent x over denominator 1 plus x to the power of 3 end exponent end fraction$ is

For such questions, we should know different methods to solve differential equation.

The solution of $left parenthesis x minus y to the power of 3 end exponent right parenthesis d x plus 3 x y to the power of 2 end exponent d y equals 0$ is

$y equals fraction numerator x over denominator x plus 1 end fraction$ is a solution of the differential equation —.

For such questions, we should know different method to solve differential equation.

$y equals fraction numerator x over denominator x plus 1 end fraction$ is a solution of the differential equation —.

For such questions, we should know different method to solve differential equation.

The degree of the differential equation $y left parenthesis x right parenthesis equals 1 plus fraction numerator d y over denominator d x end fraction plus fraction numerator 1 over denominator 1.2 end fraction left parenthesis fraction numerator d y over denominator d x end fraction right parenthesis to the power of 2 end exponent plus fraction numerator 1 over denominator 1.2.3 end fraction left parenthesis fraction numerator d y over denominator d x end fraction right parenthesis to the power of 3 end exponent plus$ is

For such questions, we should know the definition of power.

The degree of the differential equation $y left parenthesis x right parenthesis equals 1 plus fraction numerator d y over denominator d x end fraction plus fraction numerator 1 over denominator 1.2 end fraction left parenthesis fraction numerator d y over denominator d x end fraction right parenthesis to the power of 2 end exponent plus fraction numerator 1 over denominator 1.2.3 end fraction left parenthesis fraction numerator d y over denominator d x end fraction right parenthesis to the power of 3 end exponent plus$ is

For such questions, we should know the definition of power.

If $left parenthesis fraction numerator 2 plus blank s i n blank x over denominator 1 plus y end fraction right parenthesis fraction numerator d y over denominator d x end fraction equals negative blank c o s blank x comma$ $y left parenthesis 0 right parenthesis equals 1$ , then $y left parenthesis fraction numerator pi over denominator 2 end fraction right parenthesis equals$

The differential equation $fraction numerator d to the power of 2 end exponent y over denominator 2 end fraction plus x fraction numerator d y over denominator d x end fraction plus blank s i n blank y plus x to the power of 2 end exponent equals 0$ is of the $d x$ following type

The point of intersection of tangents at the ends of the latus rectum of the parabola $y to the power of 2 end exponent equals 4 x text is equal to end text$