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

If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.

Differentiate the function with respect to x:

x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.

If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.

Integrate the function `(3x^2)/(x^6 + 1)`

Integrate the function `1/sqrt(1+4x^2)`

Integrate the function `1/sqrt((2-x)^2 + 1)`

Integrate the function `1/sqrt(9 - 25x^2)`

Integrate the function `(3x)/(1+ 2x^4)`

Integrate the function `x^2/(1 - x^6)`

Integrate the function `(x - 1)/sqrt(x^2 - 1)`

Integrate the function `x^2/sqrt(x^6 + a^6)`

Integrate the function `(sec^2 x)/sqrt(tan^2 x + 4)`

Integrate the function `1/sqrt(x^2 +2x + 2)`

Integrate the function `1/(9x^2 + 6x + 5)`

Integrate the function `1/sqrt(7 - 6x - x^2)`

Integrate the function `1/sqrt((x -1)(x - 2))`

Integrate the function `1/sqrt(8+3x  - x^2)`

Integrate the function `1/sqrt((x - a)(x - b))`