Commerce (English Medium) Class 12 - CBSE Important Questions

If `d/dx f(x) = 2x + 3/x` and f(1) = 1, then f(x) is ______.

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

The value of `int_0^(π/4) (sin 2x)dx` is ______.

Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.

Find: `int e^(x^2) (x^5 + 2x^3)dx`.

Evaluate: `int_0^π x/(1 + sinx)dx`.

For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x  dx` is ______.

Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.

Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.

Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.

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

Using integration, find the area of the region {(x, y) : x² + y² ≤ 1 ≤ x + y}.

Find the area of the smaller region bounded by the ellipse \[\frac{x^2}{9} + \frac{y^2}{4} = 1\] and the line \[\frac{x}{3} + \frac{y}{2} = 1 .\]

Find the area of the region. 

{(x,y) : 0 ≤ y ≤ x² , 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .

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)`.

Find the particular solution of the differential equation  `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0

Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`

Show that the differential equation 2y^x/y dx + (y − 2x e^x/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Solve the differential equation :

`y+x dy/dx=x−y dy/dx`