Arts (English Medium) Class 12 - CBSE Important Questions for Mathematics

Evaluate: \[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx\] .

Evaluate : \[\int\frac{dx}{\sin^2 x \cos^2 x}\] .

Evaluate : \[\int\limits_0^\frac{\pi}{4} \tan x dx\] .

Evaluate : \[\int\frac{x \cos^{- 1} x}{\sqrt{1 - x^2}}dx\] .

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

Evaluate : \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \tan x}dx\] .

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))` .   

Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.

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")`

Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration. 

Evaluate: `int_-π^π (1 - "x"^2) sin "x" cos^2 "x"  d"x"`.

Evaluate:  `int_-1^2 (|"x"|)/"x"d"x"`.

Find: `int_  (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`

Evaluate: `int_1^5{|"x"-1|+|"x"-2|+|"x"-3|}d"x"`.