Science (English Medium) इयत्ता १२ - CBSE Important Questions for Mathematics

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

Find `int (cos theta)/((4 + sin^2 theta)(5 - 4 cos^2 theta)) d theta`

Find `int (sin^2 x - cos^2x)/(sin x cos x) dx`

Find `int dx/(5 - 8x - x^2)`

Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`

Find `int (2x)/(x^2 + 1)(x^2 + 2)^2 dx`

Find `int (2x)/((x^2 + 1)(x^4 + 4))`dx

Find `int((3 sin x - 2) cos x)/(13 - cos^2 x- 7 sin x) dx`

Evaluate the following integral:

Evaluate each of the following integral:

Find : `∫_a^b logx/x` dx

Find : 

`∫ sin(x-a)/sin(x+a)dx`

Find : 

`∫(log x)^2 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))` .   

Find `int_  (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `

Find `int_  sin ("x" - a)/(sin ("x" + a )) d"x"`

Find `int_  (log "x")^2 d"x"`

Prove that `int_a^b ƒ ("x") d"x" = int_a^bƒ(a + b - "x") d"x" and "hence evaluate" int_(π/6)^(π/3) (d"x")/(1+sqrt(tan "x")`