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Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Concept: Methods of Integration> Integration Using Partial Fraction
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Concept: Methods of Integration> Integration by Substitution
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Concept: Properties of Definite Integrals
Using integration, find the area of the region {(x, y) : x2 + y2 ≤ 1 ≤ x + y}.
Concept: Area Under Simple Curves
Find the area of the smaller region bounded by the ellipse \[\frac{x^2}{9} + \frac{y^2}{4} = 1\] and the line \[\frac{x}{3} + \frac{y}{2} = 1 .\]
Concept: Area Under Simple Curves
Find the area of the region.
{(x,y) : 0 ≤ y ≤ x2 , 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .
Concept: Area Under Simple Curves
Using integration find the area of the triangle formed by negative x-axis and tangent and normal to the circle `"x"^2 + "y"^2 = 9 "at" (-1,2sqrt2)`.
Concept: Area Under Simple Curves
Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`
Concept: Order and Degree of a Differential Equation
Show that the differential equation 2yx/y dx + (y − 2x ex/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.
Concept: Forms of Solving Differential Equations> Homogeneous Differential Equations
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
Concept: Forms of Solving Differential Equations> Homogeneous Differential Equations
Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.
Concept: Forms of Solving Differential Equations> Homogeneous Differential Equations
Which of the following is a homogeneous differential equation?
Concept: Forms of Solving Differential Equations> Homogeneous Differential Equations
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Concept: Basic Concepts of Differential Equations
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
Concept: Basic Concepts of Differential Equations
Write the sum of the order and degree of the differential equation
\[\left( \frac{d^2 y}{{dx}^2} \right)^2 + \left( \frac{dy}{dx} \right)^3 + x^4 = 0 .\]
Concept: Order and Degree of a Differential Equation
Solve the following differential equation : \[\left[ y - x \cos\left( \frac{y}{x} \right) \right]dy + \left[ y \cos\left( \frac{y}{x} \right) - 2x \sin\left( \frac{y}{x} \right) \right]dx = 0\] .
Concept: Forms of Solving Differential Equations> Homogeneous Differential Equations
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Concept: Basic Concepts of Differential Equations
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Concept: Basic Concepts of Differential Equations
Write the order and degree of the differential equation `((d^4"y")/(d"x"^4))^2 = [ "x" + ((d"y")/(d"x"))^2]^3`.
Concept: Order and Degree of a Differential Equation
Write the order and the degree of the following differential equation: `"x"^3 ((d^2"y")/(d"x"^2))^2 + "x" ((d"y")/(d"x"))^4 = 0`
Concept: Order and Degree of a Differential Equation
