Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

Find `int x^2/(x^4 + 3x^2 + 2) "d"x`

Evaluate `int "dx"/sqrt((x - alpha)(beta - x)), beta > alpha`

Find `int sqrt(10 - 4x + 4x^2)  "d"x`

Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`

Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`

If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4e^x + 5e –x| + C, then ______.

If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.

Verify the following:

`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`

Verify the following:

`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`

Evaluate the following:

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

`int (cos2x - cos 2theta)/(cosx - costheta) "d"x` is equal to ______.

`int "e"^x ((1 - x)/(1 + x^2))^2  "d"x` is equal to ______.

`int x^9/(4x^2 + 1)^6  "d"x` is equal to ______.

`int x^3/(x + 1)` is equal to ______.

If `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`, then ______.

`int (x + 3)/(x + 4)^2 "e"^x  "d"x` = ______.

The angle between two vectors `vec"a"` and `vec"b"` with magnitudes `sqrt(3)` and 4, respectively, and `vec"a" * vec"b" = 2sqrt(3)` is ______.

The vectors `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"` and `vec"b" = -hat"i" - 2hat"k"` are the adjacent sides of a parallelogram. The acute angle between its diagonals is ______.

The derivative of `"cos"^-1 ((1 - "x"^2)/(1 + "x"^2))` with respect to `"cot"^-1 ((1 - 3"x"^2)/(3"x" - "x"^3))` is ____________.

The derivative of `"sin"^-1 ((2"x")/(1 + "x"^2))` with respect to `"tan"^-1 ((2"x")/(1 - "x"^2))` is ____________.