Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

The derivative of log₁₀x w.r.t. x is ______.

If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.

If y^x = e^{y – x}, prove that `"dy"/"dx" = (1 + log y)^2/logy`

If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`

Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`

Verify the following using the concept of integration as an antiderivative

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

Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x² = t

Evaluate the following:

`int (x^2"d"x)/(x^4 - x^2 - 12)`

Evaluate the following:

`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`

Evaluate the following:

`int_"0"^pi  (x"d"x)/(1 + sin x)`

Evaluate the following:

`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`

Evaluate the following:

`int "e"^(-3x) cos^3x  "d"x`

Evaluate the following:

`int sqrt(tanx)  "d"x`  (Hint: Put tanx = t²)

If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.

Find the solution of `"dy"/"dx"` = 2^y–x.

Find the differential equation of all non-vertical lines in a plane.

Solve the differential equation `(x^2 - 1) "dy"/"dx" + 2xy = 1/(x^2 - 1)`.

Solve the differential equation `"dy"/"dx" + 1` = e^{x + y}.

The determinant `abs (("a","bc","a"("b + c")),("b","ac","b"("c + a")),("c","ab","c"("a + b"))) =` ____________

Solve: (x + y)(dx – dy) = dx + dy. [Hint: Substitute x + y = z after seperating dx and dy]