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Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Concept: Methods of Integration> Integration Using Partial Fraction
Evaluate `int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx`
Concept: Evaluation of Definite Integrals by Substitution
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Concept: Methods of Integration> Integration by Parts
Find: `int (dx)/(x^2 - 6x + 13)`
Concept: Integrals of Some Particular Functions
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
Concept: Properties of Definite Integrals
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
Concept: Integration as an Inverse Process of Differentiation
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Concept: Methods of Integration> Integration by Substitution
`int secx/(secx - tanx)dx` equals ______.
Concept: Methods of Integration> Integration by Substitution
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Concept: Properties of Definite Integrals
Evaluate: `int_0^(π/2) sin 2x tan^-1 (sin x) dx`.
Concept: Evaluation of Definite Integrals by Substitution
Using integration find the area of the region {(x, y) : x2+y2⩽ 2ax, y2⩾ ax, x, y ⩾ 0}.
Concept: Area Under Simple Curves
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
`x^2+y^2=4 at (1, sqrt3)`
Concept: Area Under Simple Curves
Find the area bounded by the circle x2 + y2 = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.
Concept: Area Under Simple Curves
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Concept: General and Particular Solutions of a Differential Equation
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Concept: General and Particular Solutions of a Differential Equation
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Concept: General and Particular Solutions of a Differential Equation
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Concept: General and Particular Solutions of a Differential Equation
Find the general solution of the following differential equation:
`(dy)/(dx) = e^(x-y) + x^2e^-y`
Concept: Order and Degree of a Differential Equation
Read the following passage:
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An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
Based on the above, answer the following questions:
- Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
- Solve the above equation to find its general solution. (2)
Concept: Methods of Solving Differential Equations> Homogeneous Differential Equations
Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1
Concept: Basic Concepts of Vector Algebra
