Arts (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

`int e^x sec x (1 +   tan x) dx` equals:

Find the area of a parallelogram whose adjacent sides are represented by the vectors\[2 \hat{i} - 3 \hat{k} \text { and } 4 \hat{j} + 2 \hat{k} .\]

Find : 

`∫(log x)^2 dx`

If x^y - y^x = a^b, find `(dy)/(dx)`.

If `"x" = "e"^(cos2"t")  "and"  "y" = "e"^(sin2"t")`, prove that `(d"y")/(d"x") = - ("y"log"x")/("x"log"y")`.

If x^y = e^x–y, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`

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)`

Find `int_0^1 x(tan^-1x)  "d"x`

Evaluate the following:

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

Evaluate the following:

`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`

Evaluate the following:

`int_0^1 x log(1 + 2x)  "d"x`

Evaluate the following:

`int_0^pi x log sin x "d"x`

`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.

`int tan^-1 sqrt(x)  "d"x` is equal to ______.

The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is ______.

If `"y" = ("x" + sqrt(1 + "x"^2))^"n",  "then" (1 + "x"^2)  ("d"^2 "y")/"dx"^2 + "x" ("dy")/("dx")` is ____________.