HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

State whether the following statement is True or False:

If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1|  + B log|x – 2|, then A + B = 1

State whether the following statement is True or False:

For `int (x - 1)/(x + 1)^3  "e"^x"d"x` = e^x f(x) + c, f(x) = (x + 1)²

State whether the following statement is True or False:

`int sqrt(1 + x^2) *x  "d"x = 1/3(1 + x^2)^(3/2) + "c"`

Evaluate `int (2x + 1)/((x + 1)(x - 2))  "d"x`

`int x/((x - 1)^2 (x + 2)) "d"x`

`int 1/sqrt(x^2 - 9) dx` = ______.

State whether the following statement is true or false.

If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2e^x – 5| + c, where c is the constant of integration, then A = 5.

`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`

`int (f^'(x))/(f(x))dx` = ______ + c.

`int(7x - 2)^2dx = (7x -2)^3/21 + c`

`int 1/sqrt(x^2 - a^2)dx` = ______.

Solve: `int sqrt(4x^2 + 5)dx`

`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`

The value of `int ("d"x)/(sqrt(1 - x))` is ______.

Complete the following activity:

`int_0^2 dx/(4 + x - x^2) `

= `int_0^2 dx/(-x^2 + square + square)`

= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`

= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`

= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`

Find the area of the region bounded by the parabola y² = 16x and the line x = 4. 

Find the area of the ellipse `x^2/4 + y^2/25 = 1`

Find the area of the region bounded by the following curves, the X-axis and the given lines:

y = x² + 1, x = 0, x = 3

Area of the region bounded by y = x⁴, x = 1, x = 5 and the X-axis is _______.

Using definite integration, area of the circle x² + y² = 49 is _______.