Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

By using the properties of the definite integral, evaluate the integral:

`int_0^a  sqrtx/(sqrtx + sqrt(a-x))   dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^4 |x - 1| dx`

Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx`  if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.

`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.

The value of `int_0^(pi/2) log  ((4+ 3sinx)/(4+3cosx))` dx is ______.

Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`

Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`

\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.

\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.

If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that

Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .

Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx`  and hence evaluate   `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .   

Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`

Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.

Find : `int_  (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.

Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`

Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`

Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`

Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`