∫(logx)2dx equals ______. - Mathematics (JEE Main)

Advertisements
Advertisements
MCQ
Fill in the Blanks

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

Options

  • (x logx)2 – 2x logx + 2x + c

  • x(logx)2 – 2x logx + 2x + c

  • x(logx)2 + 2x logx + 2x + c

  • x(logx)2 + 2x logx – 2x + c

Advertisements

Solution

`int(logx)^2dx` equals `underlinebb(x(log x)^2 - 2x log x + 2x + c)`.

Explanation:

I = `int1.(logx)^2dx`  ...[Using by parts]

Take 1 as the second function then,

I = `x(logx)^2 - intx.(2logx)/xdx`

= `x(logx)^2 - 2intlogxdx`

Again by parts

I = `x(logx)^2 - 2[xlogx - intx. 1/x dx]`

= x(logx)2 – 2x logx + 2x + c

  Is there an error in this question or solution?

RELATED QUESTIONS

Prove that:

`int sqrt(a^2 - x^2) dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a)+c`


Integrate : sec3 x w. r. t. x.


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`


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

(A)log(log x)+ c

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

(C) 2log x + c

(D) log x + c


If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`


Integrate the function in x sin x.


Integrate the function in `x^2e^x`.


Integrate the function in x log x.


Integrate the function in x log 2x.


Integrate the function in xlog x.


Integrate the function in x sin-1 x.


Integrate the function in x cos-1 x.


Integrate the function in (sin-1x)2.


Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.


Integrate the function in x sec2 x.


Integrate the function in tan-1 x.


Integrate the function in x (log x)2.


Integrate the function in `(xe^x)/(1+x)^2`.


Integrate the function in `e^x (1 + sin x)/(1+cos x)`.


Integrate the function in e2x sin x.


Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.


Find : 

`∫(log x)^2 dx`


Evaluate the following : `int x^2 sin 3x  dx`


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


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


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


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


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


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 : `sqrt(2x^2 + 3x + 4)`


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 : `log(1 + x)^((1 + 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 cos -(3)/(7)x*sin -(11)/(7)x*dx` =


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


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 (log x)+(log x)–2 


Integrate the following w.r.t.x : e2x sin x cos x


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


Evaluate the following.

`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` 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: `int "e"^"x"/(4"e"^"2x" -1)` dx


`int 1/(4x + 5x^(-11))  "d"x`


`int 1/sqrt(2x^2 - 5)  "d"x`


`int (cos2x)/(sin^2x cos^2x)  "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)` = ______


Choose the correct alternative:

`int ("d"x)/((x - 8)(x + 7))` =


`int 1/x  "d"x` = ______ + c


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


Evaluate `int 1/(4x^2 - 1)  "d"x`


`int logx/(1 + logx)^2  "d"x`


`int 1/sqrt(x^2 - 8x - 20)  "d"x`


`int [(log x - 1)/(1 + (log x)^2)]^2`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 = ______.


Evaluate the following:

`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`


Evaluate the following:

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


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` = ______.


Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`


Find: `int e^x.sin2xdx`


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 e^x [(2 + sin 2x)/(1 + cos 2x)]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 ((1 - sinx)/(1 - cosx))dx`.


`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`


The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.


`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`


Solve the following

`int_0^1 e^(x^2) x^3 dx`


Evaluate:

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


Evaluate:

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


Evaluate:

`inte^x sinx  dx`


Evaluate:

`int e^(logcosx)dx`


Evaluate:

`int (logx)^2 dx`


`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))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`


Evaluate:

`int (sin(x - a))/(sin(x + a))dx`


Complete the following activity:

`int_0^2 dx/(4 + x - x^2) `

= `int_0^2 dx/(-x^2 + square + square)`

= `int_0^2 dx/ -x^2 + x + 1/4 - square + 4`

= `int_0^2 dx/ ((x- 1/2) - (square)^2)`

= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`


Evaluate: `int1/(x^2 + 25)dx`


Evaluate the following:

`intx^3e^(x^2)dx` 


Evaluate the following.

`intx^3 e^(x^2) dx`


Evaluate the following.

`intx^3/sqrt(1+x^4)  dx`


Evaluate the following.

`intx^3e^(x^2) dx`


Evaluate `int (1 + x + x^2/(2!))dx`


Share
Notifications



      Forgot password?
Use app×