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

Prove that `(veca + vecb).(veca + vecb)` = `|veca|^2 + |vecb|^2` if and only if `veca . vecb` are perpendicular, given `veca != vec0, vecb != vec0.`

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

`int_0^(pi/2) cos^2 x dx`

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

`int_0^(pi/2)  sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx` 

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

`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`

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

`int_0^(pi/2)  (cos^5  xdx)/(sin^5 x + cos^5 x)`

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

`int_(-5)^5 | x + 2| dx`

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

`int_2^8 |x - 5| dx`

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

`int_0^1 x(1-x)^n dx`

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

`int_0^(pi/4) log (1+ tan x) dx`

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

`int_0^2 xsqrt(2 -x)dx`

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

`int_0^(pi/2) (2log sin x - log sin 2x)dx`

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

`int_((-pi)/2)^(pi/2) sin^2 x  dx`

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

`int_0^pi (x  dx)/(1+ sin x)`

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

`int_(pi/2)^(pi/2) sin^7 x dx`

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

`int_0^(2x) cos^5 xdx`

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

`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`

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

`int_0^pi log(1+ cos x) dx`

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.