HSC Science (Electronics) १२ वीं कक्षा - Maharashtra State Board Important Questions

If u and v ore differentiable functions of x. then prove that:

`int uv  dx = u intv  dx - int [(du)/(d) intv  dx]dx`

Hence evaluate `intlog x  dx`

`int cos^3x  dx` = ______.

Write `int cotx  dx`.

Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`

Evaluate:

`int1/(x^2 + 25)dx`

Evaluate the following integrals as limit of a sum:

\[\int\limits_0^2 (3x^2 - 1)\cdot dx\]

Evaluate: `int_0^1 (x^2 - 2)/(x^2 + 1).dx`

Evaluate: `int_0^(pi/2) x sin x.dx`

Evaluate: `int_0^π sin^3x (1 + 2cosx)(1 + cosx)^2.dx`

Choose the correct option from the given alternatives : 

`int_0^(pi/2) (sin^2x*dx)/(1 + cosx)^2` = ______.

If `int_0^1 ("d"x)/(sqrt(1 + x) - sqrt(x)) = "k"/3`, then k is equal to ______.

Let I₁ = `int_"e"^("e"^2)  1/logx  "d"x` and I₂ = `int_1^2 ("e"^x)/x  "d"x` then 

`int_0^(pi/2) log(tanx)  "d"x` =

Evaluate: `int_(pi/6)^(pi/3) cosx  "d"x`

Evaluate: `int_0^1 "e"^x/sqrt("e"^x - 1)  "d"x`

Evaluate: `int_1^3 (cos(logx))/x  "d"x`

Evaluate: `int_0^1 (1/(1 + x^2)) sin^-1 ((2x)/(1 + x^2))  "d"x`

Evaluate: `int_0^pi 1/(3 + 2sinx + cosx)  "d"x`

Evaluate: `int_0^(π/4) sec^4 x  dx`

If `int_2^e [1/logx - 1/(logx)^2].dx = a + b/log2`, then ______.