Balbharati solutions for माठेमटिक्स अँड स्टॅटिस्टिक्स २ (आर्ट्स अँड सायन्स) [इंग्रजी] इयत्ता १२ महाराष्ट्र राज्य मंडळ chapter 4 - Definite Integration [Latest edition]

Evaluate the following integrals as limit of a sum : `""int_1^3 (3x - 4).dx`

Evaluate the following integrals as limit of a sum:

\[\int\limits_0^4 x^2 \cdot dx\]

Evaluate the following integrals as limit of a sum:

`int _0^2 e^x * dx`

Evaluate the following integrals as limit of a sum:

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

Evaluate the following integrals as limit of a sum : \[\int\limits_1^3 x^3 \cdot dx\]

Evaluate : `int_1^9(x + 1)/sqrt(x)*dx`

Evaluate : `int_2^3 (1)/(x^2 + 5x + 6)*dx`

Evaluate: `int_0^(π/4) cot^2x.dx`

Evaluate:

`int_(-pi/4)^(pi/4) (1)/(1 - sinx)*dx`

Evaluate : `int_3^5 (1)/(sqrt(2x + 3) - sqrt(2x - 3))*dx`

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

Evaluate : `int_0^(pi/4) sin 4x sin 3x *dx`

Evaluate:

`int_0^(pi/4) sqrt(1 + sin 2x)*dx`

Evaluate : `int_0^(pi/4) sin^4x*dx`

Evaluate: `int_(-4)^2 (1)/(x^2 + 4x + 13)*dx`

Evaluate : `int_0^4 (1)/sqrt(4x - x^2)*dx`

Evaluate:

`int_0^1 (1)/sqrt(3 + 2x - x^2)*dx`

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

Evaluate:

`int_0^1 x tan^-1x*dx`

Evaluate: `int_0^oo xe^-x.dx`

Evaluate:

`int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2)*dx`

Evaluate : `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x +1)*dx`

Evaluate : `int_0^(pi//4) (sin2x)/(sin^4x + cos^4x)*dx`

Evaluate:

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

Evaluate : `int_0^(pi/2) (1)/(5 + 4 cos x)*dx`

Evaluate : `int_0^(pi/4) (cosx)/(4 - sin^2x)*dx`

Evaluate : `int_0^(pi/2) cosx/((1 + sinx)(2 + sin x))*dx`

Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`

Evaluate : `int_0^pi (1)/(3 + 2sinx + cosx)*dx`

Evaluate : `int_0^(pi/4) sec^4x*dx`

Evaluate:

`int_0^1 sqrt((1 - x)/(1 + x)) * dx`

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

Evaluate: `int_0^(pi/2) sin2x*tan^-1 (sinx)*dx`

Evaluate : `int _((1)/(sqrt(2)))^1 (e^(cos^-1x) sin^-1x)/(sqrt(1 - x^2))*dx`

Evaluate : `int_1^3 (cos(logx))/x*dx`

Evaluate the following:

`int_0^a (1)/(x + sqrt(a^2 - x^2)).dx`

Evaluate the following:

`int_0^(pi/2) log(tanx)dx`

Evaluate the following integrals : `int_0^1 log(1/x - 1)*dx`

Evaluate : `int_0^(pi/2) (sinx - cosx)/(1 + sinx cosx)*dx`

Evaluate the following : `int_0^3 x^2(3 - x)^(5/2)*dx`

Evaluate the following : `int_(-3)^(3) x^3/(9 - x^2)*dx`

Evaluate the following:

`int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`

Evaluate the following :  `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`

Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`

Evaluate the following: `int_0^1 (log(x + 1))/(x^2 + 1)*dx`

Evaluate the following : `int_(-1)^(1) (x^3 + 2)/sqrt(x^2 + 4)*dx`

Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`

Evaluate the following : `int_0^1 t^2 sqrt(1 - t)*dt`

Evaluate the following : `int_0^pi x sin x cos^2x*dx`

Evaluate the following : `int_0^1 (logx)/sqrt(1 - x^2)*dx`

`int_2^3 dx/(x(x^3 - 1))` = ______.

Choose the correct option from the given alternatives : 

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

`int_0^(log5) (e^x sqrt(e^x - 1))/(e^x + 3) * dx` = ______.

Choose the correct option from the given alternatives : 

`int_0^(pi/2) sn^6x cos^2x*dx` =

If `dx/(sqrt(1 + x) − sqrt(x)) = k/(3)`, then k is equal to ______.

`int_1^2 (1)/(x^2) * e^(1/x)  dx` = ______.

Choose the correct option from the given alternatives :

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

Choose the correct option from the given alternatives :

Let I₁ = `int_e^(e^2) dx/logx  "and"  "I"_2 = int_1^2 e^x/x*dx`, then

Choose the correct option from the given alternatives :

`int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =

Choose the correct option from the given alternatives :

The value of `int_((-pi)/4)^(pi/4) log((2+ sin theta)/(2 - sin theta))*d theta` is

Evaluate the following : `int_0^(pi/2) cosx/(3cosx + sinx)*dx`

Evaluate the following : `int_(pi/4)^(pi/2) (cos theta)/[cos  theta/2 + sin  theta/2]^3*d theta`

Evaluate the following : `int_0^1 1/(1 + sqrt(x))*dx`

Evaluate the following : `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`

Evaluate the following : `int_0^1 t^5 sqrt(1 - t^2)*dt`

Evaluate the following : `int_0^1 (cos^-1 x^2)*dx`

Evaluate the following : `int_(-1)^(1) (1 + x^3)/(9 - x^2)*dx`

Evaluate the following : `int_0^pi x*sinx*cos^4x*dx`

Evaluate the following:

`int_0^pi x/(1 + sin^2x) * dx`

Evaluate the following : `int_1^oo 1/(sqrt(x)(1 + x))*dx`

Evaluate the following : `int_0^1 (1/(1 + x^2))sin^-1((2x)/(1 + x^2))*dx`

Evaluate the following : `int_0^(pi/2) 1/(6 - cosx)*dx`

Evaluate the following : `int_0^a 1/(a^2 + ax - x^2)*dx`

Evaluate the following : `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx`

Evaluate the following : `int_0^1 sin^-1 ((2x)/(1 + x^2))*dx`

Evaluate the following : `int_0^(pi/4) (cos2x)/(1 + cos 2x + sin 2x)*dx`

Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`

Evaluate the following : `int_0^pi  (sin^-1x + cos^-1x)^3 sin^3x*dx`

Evaluate the following : `int_0^4 [sqrt(x^2 + 2x + 3]]^-1*dx`

Evaluate the following : `int_(-2)^(3) |x - 2|*dx`

Evaluate the following : if `int_a^a sqrt(x)*dx = 2a int_0^(pi/2) sin^3x*dx`, find the value of `int_a^(a + 1)x*dx`

Evaluate the following : If `int_0^k 1/(2 + 8x^2)*dx = pi/(16)`, find k

Evaluate the following : If f(x) = a + bx + cx², show that `int_0^1 f(x)*dx = (1/(6)[f(0) + 4f(1/2) + f(1)]`