Integrate the following w.r.t.x : cot–1 (1 – x + x2) - Mathematics and Statistics

Advertisements
Advertisements
Sum

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

Advertisements

Solution

Let I = `int cot^-1 (1 - x + x^2)*dx`

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

= `int tan^-1 [(x + (1 - x))/(1 - x(1 - x))]`

= `int [tan^-1 x + tan^-1 (1 - x)]*dx`

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

∴ I = I1 + I2                                                 ...(1)

I1 = `int tan^-1 x*dx = int(tan^-1x)1*dx`

= `(tan^-1x)* int 1dx - [d/dx (tan^-1x)* int 1dx]*dx`

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

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

∴ I1 = `x tan^-1x - (1)/(2)log|1 + x^2| + c_1`

  ...`[because d/dx (1 + x^2) = 2x and int (f'(x))/f(x) dx = log|f(x)| + c]`

I2 = `int tan^-1 (1 - x)*dx`

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

= `[tan^-1 (1 - x)]*int 1dx - int {d/dx [tan^-1 (1 - x)]* int 1dx}*dx`

= `[tan^-1 (1 - x)]*x - int (1)/(1 + (1 - x)^2)*(-1)*xdx`

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

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

Let x = `"A"[d/dx (2 - 2x + x^2)] + "B"`

∴ x = A(– 2 + 2x) + B = 2Ax + (–2A + B)
Comparing the coefficient of x and constant on both the sides, we get
1 = 2A and 0 = – 2A + B

∴ A = `(1)/(2) and 0 = -2(1/2) + "B"`

∴ B = 1

∴ x = `(1)/(2)(- 2 + 2x) + 1`

∴ I2= `xtan^-1 (1 - x) + int (1/2(-2 + 2x) + 1)/(2 - 2x + x^2)*dx`

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

= `xtan^-1 (1 - x) + (1)/(2) log|2 - 2x + x^2| + int (1)/(1 + (1 - 2x + x^2))*dx`

= `xtan^-1 (1 - x) + (1)/(2) log|x^2 - 2x + 2| + int (1)/(1 + (1 -  x^2))*dx`

= `xtan^-1 (1 - x) + (1)/(2) log|x^2 - 2x + 2| + (1)/(1) (tan-1 (1 - x))/(-1) + c_2`

= `x tan^-1 (1 - x) + 1/2log|x^2 - 2x + 2| - tan^-1 (1 - x) + c_2`

= `(x - 1)tan^-1 (1 - x) + (1)/(2)log|x^2 - 2x + 2| + c_2`

∴ I2 = `-(1 - x)tan^-1 (1 - x) + (1)/(2)log|x^2 - 2x + 2| + c_2`             ...(3)

From (1),(2) and (3), we get

I = `x tan^-1 x - (1)/(2) log|1 + x^2| + c_1 - (1 - x)tan^-1 (1 - x) + 1/2log|x^2 - 2x + 2| + c_2`

= `x tan^-1 x - (1)/(2) log|1 + x^2| - (1 - x)tan^-1 (1 - x) + 1/2 |x^2 - 2x + 2| + c`, where c = c1 + c2.

  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Miscellaneous Exercise 3 [Page 150]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 3.02 | Page 150

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Prove that:

`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`


If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:

(A) 0

(B) π

(C) π/2

(D) π/4


`int1/xlogxdx=...............`

(A)log(log x)+ c

(B) 1/2 (logx )2+c

(C) 2log x + c

(D) log x + c


Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`


Integrate the functions x sin x


Integrate the functions x sin 3x


Integrate the functions x logx


Integrate the functions xlog x


Integrate the functions x sin– 1x


Integrate the functions x tan–1 x


Integrate the functions x cos–1 x


Integrate the functions (sin–1x)2


Integrate the functions x (log x)2


Integrate the functions (x2 + 1) log x


Integrate the functions ex (sinx + cosx)


Integrate the functions `((x- 3)e^x)/(x - 1)^3`


Integrate the functions e2x sin x


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


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


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


Evaluate the following : `int x^2*cos^-1 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 : `sqrt(5x^2 + 3)`


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 : `sec^2x.sqrt(tan^2x + tan x - 7)`


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


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^(sin^-1x).[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`


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


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 tan(sin^-1 x)*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 :

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


Choose the correct options from the given alternatives :

`int sin (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 :

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


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 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 : log (log x)+(log x)–2 


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


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


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


Evaluate the following.

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


Evaluate the following.

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


Evaluate the following.

`int "e"^"x" "x - 1"/("x + 1")^3` dx


Evaluate the following.

`int [1/(log "x") - 1/(log "x")^2]` dx


Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx


Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`


Evaluate: `int "dx"/("9x"^2 - 25)`


Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx


Evaluate: ∫ (log x)2 dx


`int (sinx)/(1 + sin x)  "d"x`


`int (sin(x - "a"))/(cos (x + "b"))  "d"x`


`int 1/sqrt(2x^2 - 5)  "d"x`


`int ["cosec"(logx)][1 - cot(logx)]  "d"x`


`int ("e"^xlog(sin"e"^x))/(tan"e"^x)  "d"x`


`int sqrt(tanx) + sqrt(cotx)  "d"x`


`int ("d"x)/(x - x^2)` = ______


`int 1/(x^2 - "a"^2)  "d"x` = ______ + c


`int (x^2 + x - 6)/((x - 2)(x - 1))  "d"x` = x + ______ + c


State whether the following statement is True or False:

If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1|  + B log|x – 2|, then A + B = 1


Evaluate `int 1/(x(x - 1))  "d"x`


Evaluate `int 1/(4x^2 - 1)  "d"x`


Evaluate `int (2x + 1)/((x + 1)(x - 2))  "d"x`


`int logx/(1 + logx)^2  "d"x`


`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?


`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.


`int cot "x".log [log (sin "x")] "dx"` = ____________.


`int log x * [log ("e"x)]^-2` dx = ?


The value of `int "e"^(5x) (1/x - 1/(5x^2))  "d"x` is ______.


`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.


`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.


Find `int_0^1 x(tan^-1x)  "d"x`


Evaluate the following:

`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`


Evaluate the following:

`int_0^1 x log(1 + 2x)  "d"x`


Evaluate the following:

`int_0^pi x log sin x "d"x`


`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.


`int tan^-1 sqrt(x)  "d"x` is equal to ______.


The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1)  dx` is


The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x))  dx` is


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 1/sqrt(x^2 - 9) dx` = ______.


State whether the following statement is true or false.

If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.


`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`


Find: `int e^x.sin2xdx`


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


`int(logx)^2dx` equals ______.


If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.


If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.


The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.


If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.


`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.


If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.


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


Find: `int e^(x^2) (x^5 + 2x^3)dx`.


Solution of the equation `xdy/dx=y log y` is ______


Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.

Solution: (x2 + y2) dx - 2xy dy = 0

∴ `dy/dx=(x^2+y^2)/(2xy)`                      ...(1)

Puty = vx

∴ `dy/dx=square`

∴ equation (1) becomes

`x(dv)/dx = square`

∴ `square  dv = dx/x`

On integrating, we get

`int(2v)/(1-v^2) dv =intdx/x`

∴ `-log|1-v^2|=log|x|+c_1`

∴ `log|x| + log|1-v^2|=logc       ...["where" - c_1 = log c]`

∴ x(1 - v2) = c

By putting the value of v, the general solution of the D.E. is `square`= cx


`int(xe^x)/((1+x)^2)  dx` = ______


Evaluate:

`intcos^-1(sqrt(x))dx`


Evaluate:

`int((1 + sinx)/(1 + cosx))e^x dx`


Evaluate:

`int e^(ax)*cos(bx + c)dx`


Evaluate:

`int e^(logcosx)dx`


Evaluate:

`int (logx)^2 dx`


Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`


The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.


Evaluate `int tan^-1x  dx`


If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv  dx - int(d/dx u)(intv  dx)dx`. Hence evaluate: `intx cos x  dx`


Evaluate: `int1/(x^2 + 25)dx`


Evaluate the following.

`intx^3/sqrt(1+x^4)  dx`


If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).


Share
Notifications



      Forgot password?
Use app×