Science (English Medium) Class 12 - CBSE Important Questions

If `tan^(-1)  (x- 3)/(x - 4) + tan^(-1)  (x +3)/(x + 4) = pi/4`, then find the value of x.

Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`

Evaluate:

\[\int \cos^{-1} \left(\sin x \right) \text{dx}\]

Find : ` int  (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`

Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`

Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.

Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`

Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`

Evaluate `int_0^(π//4) log (1 + tanx)dx`.

Find `int dx/sqrt(sin^3x cos(x - α))`.

Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.

`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (4 , 1), B (6, 6) and C (8, 4).

Find the area enclosed between the parabola 4y = 3x² and the straight line 3x - 2y + 12 = 0.

Find the particular solution of the differential equation:

2y e^x/y dx + (y - 2x e^x/y) dy = 0 given that x = 0 when y = 1.

Solve the differential equation `cos^2 x dy/dx` + y = tan x

Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.

Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x³ – 3xy²)dx = (y³ – 3x²y)dy, where C is parameter

The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.