Balbharati solutions for माठेमटिक्स अँड स्टॅटिस्टिक्स २ (आर्ट्स अँड सायन्स) [इंग्रजी] इयत्ता १२ महाराष्ट्र राज्य मंडळ chapter 3 - Indefinite Integration [Latest edition]

Integrate the following w.r.t. x : x³ + x2 – x + 1

Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`

Integrate the following w.r.t. x:

`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`

Integrate the following w.r.t. x:

`2x^3 - 5x + 3/x + 4/x^5`

Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`

Evaluate the following integrals : tan²x dx

Evaluate the following integrals : `int (sin2x)/(cosx)dx`

Evaluate the following integrals : `int sin x/cos^2x dx`

Evaluate the following integrals:

`int (cos2x)/sin^2x dx` 

Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`

Evaluate the following integrals : `int sinx/(1 + sinx)dx`

Evaluate the following integrals : `int tanx/(sec x + tan x)dx`

Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`

Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`

Evaluate the following integrals: `int sin 4x cos 3x dx`

Evaluate the following integrals:

`int x/(x + 2).dx`

Evaluate the following integral: 

`int(4x + 3)/(2x + 1).dx`

Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`

Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`

Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`

Evaluate the following integrals:

`int (sin4x)/(cos2x).dx`

Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`

Evaluate the following integrals : `int cos^2x.dx`

Evaluate the following integrals:

`int(2)/(sqrt(x) - sqrt(x + 3)).dx`

Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`

If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)

Integrate the following functions w.r.t. x : `(logx)^n/x`

Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`

Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`

Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`

Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`

Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`

Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`

Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`

Integrate the following functions w.r.t. x : sin⁴x.cos³x

Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`

Integrate the following function w.r.t. x:

x⁹.sec²(x¹⁰)

Integrate the following functions w.r.t. x : e^3logx(x⁴ + 1)^–1 

Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`

Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`

Integrate the following functions w.r.t.x:

`(2sinx cosx)/(3cos^2x + 4sin^2 x)`

Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`

Integrate the following function w.r.t. x:

`(10x^9 +10^x.log10)/(10^x + x^10)`

Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`

Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`

Integrate the following functions w.r.t. x:

`x^5sqrt(a^2 + x^2)`

Integrate the following functions w.r.t.x:

`(5 - 3x)(2 - 3x)^(-1/2)`

Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`

Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`

Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`

Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.

Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`

Integrate the following functions w.r.t. x : `cosx/sin(x - a)`

Integrate the following functions w.r.t. x : `sin(x - a)/cos(x  + b)`

Integrate the following functions w.r.t. x: 

`(1)/(sinx.cosx + 2cos^2x)`

Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`

Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`

Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`

Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`

Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`

Integrate the following functions w.r.t.x:

cos⁸xcotx

Integrate the following functions w.r.t. x : tan⁵x

Integrate the following functions w.r.t. x : cos⁷x

Integrate the following functions w.r.t. x :  tan 3x tan 2x tan x

Integrate the following functions w.r.t. x : sin⁵x.cos⁸x

Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`

Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`

Integrate the following functions w.r.t. x:

`(sinx cos^3x)/(1 + cos^2x)`

Evaluate the following : `int (1)/(4x^2 - 3).dx`

Evaluate the following:

`int (1)/(25 - 9x^2)*dx`

Evaluate the following : `int (1)/(7 + 2x^2).dx`

Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`

Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`

Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`

Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`

Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`

Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`

Evaluate the following : `int  (1)/(x^2 + 8x + 12).dx`

Evaluate the following : `int (1)/(1 + x - x^2).dx`

Evaluate the following : `(1)/(4x^2 - 20x + 17)`

Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`

Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`

Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`

Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`

Evaluate the following:

`int (1)/sqrt((x - 3)(x + 2)).dx`

Evaluate the following : `int (1)/(4 + 3cos^2x).dx`

Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`

Evaluate the following:

`int sinx/(sin 3x)  dx`

Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`

Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`

Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`

Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`

Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`

Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`

Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`

Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`

Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`

Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`

Evaluate the following integrals:

`int (2x + 1)/(x^2 + 4x - 5).dx`

Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`

Evaluate the following integrals :  `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`

Evaluate the following integrals:

`int (7x + 3)/sqrt(3 + 2x - x^2).dx`

Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`

Evaluate the following integrals : `int sqrt((9 - x)/x).dx`

Evaluate the following integral:

`int (3cosx)/(4sin^2x + 4sinx - 1).dx`

Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`

Evaluate the following : `int x^2.log x.dx`

Evaluate the following:

`int x^2 sin 3x  dx`

Evaluate the following:

`int x tan^-1 x . dx`

Evaluate the following : `int x^2tan^-1x.dx`

Evaluate the following : `int x^3.tan^-1x.dx`

Evaluate the following : `int (logx)2.dx`

Evaluate the following:

`int sec^3x.dx`

Evaluate the following : `int x.sin^2x.dx`

Evaluate the following : `int x^3.logx.dx`

Evaluate the following : `int e^(2x).cos 3x.dx`

Evaluate the following: `int x.sin^-1 x.dx`

Evaluate the following : `int x^2*cos^-1 x*dx`

Evaluate the following : `int log(logx)/x.dx`

Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`

Evaluate the following : `int cos sqrt(x).dx`

Evaluate the following : `int sin θ.log (cos θ).dθ`

Evaluate the following : `int x.cos^3x.dx`

Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`

Evaluate the following: `int logx/x.dx`

Evaluate the following:

`int x.sin 2x. cos 5x.dx`

Evaluate the following : `int cos(root(3)(x)).dx`

Integrate the following functions w.r.t. x : `e^(2x).sin3x`

Integrate the following functions w.r.t.x:

`e^-x cos2x`

Integrate the following functions w.r.t. x:

sin (log x)

Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`

Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`

Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`

Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`

Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`

Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`

Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`

Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.

Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`

Integrate the following functions w.r.t. x : [2 + cot x – cosec²x]e^x  

Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`

Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`

Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`

Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`

Integrate the following functions w.r.t.x:

`e^(5x).[(5x.logx + 1)/x]`

Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`

Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`

Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)] 

Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`

Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`

Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`

Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`

Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`

Integrate the following w.r.t. x:

`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`

Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`

Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`

Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`

Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`

Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`

Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`

Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`

Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`

Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`

Integrate the following w.r.t. x : `(1)/(x^3 - 1)`

Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`

Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`

Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`

Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`

Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`

Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`

Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`

Choose the correct option from the given alternatives : 

`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =

Choose the correct options from the given alternatives :

`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =

Choose the correct options from the given alternatives :

`int (log (3x))/(xlog (9x))*dx` =

Choose the correct options from the given alternatives :

`int (sin^m x)/(cos^(m+2)x)*dx` = 

Choose the correct options from the given alternatives :

`int tan(sin^-1 x)*dx` =

Choose the correct options from the given alternatives :

`int (x- sinx)/(1 - cosx)*dx` =

If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.

Choose the correct options from the given alternatives :

If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =

Choose the correct options from the given alternatives :

`int (1)/(cosx - cos^2x)*dx` =

Choose the correct options from the given alternatives :

`int sqrt(cotx)/(sinx*cosx)*dx` =

Choose the correct options from the given alternatives :

`int (e^x(x - 1))/x^2*dx` =

Choose the correct options from the given alternatives :

`int sin (log x)*dx` =

Choose the correct options from the given alternatives :

`int f x^x (1 + log x)*dx`

Choose the correct options from the given alternatives :

`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =

Choose the correct options from the given alternatives :

`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =

Choose the correct options from the given alternatives : 

`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =

`int logx/(log ex)^2*dx` = ______.

Choose the correct options from the given alternatives :

`int [sin (log x) + cos (log x)]*dx` =

Choose the correct options from the given alternatives :

`int (cos2x - 1)/(cos2x + 1)*dx` =

Choose the correct options from the given alternatives :

`int (e^(2x) + e^-2x)/e^x*dx` =

Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`

Integrate the following with respect to the respective variable:

`x^7/(x + 1)`

Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`

Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`

Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`

Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`

Integrate the following with respect to the respective variable : cos 3x cos 2x cos x

Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`

Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`

Integrate the following w.r.t. x: `(1 + log x)^2/x`

Integrate the following w.r.t.x : cot^–1 (1 – x + x²)

Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`

Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`

Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`

Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`

Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`

Integrate the following w.r.t.x : log (log x)+(log x)^–2 

Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`

Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`

Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`

Integrate the following w.r.t.x : log (x² + 1)

Integrate the following w.r.t.x : e^2x sin x cos x

Integrate the following w.r.t.x:

`x^2/((x - 1)(3x - 1)(3x - 2)`

Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`

Integrate the following w.r.t.x :  `sec^2x sqrt(7 + 2 tan x - tan^2 x)`

Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`

Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`

Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`

Integrate the following w.r.t.x : sec⁴x cosec²x