Evaluate the following : ∫x2.cos-1x.dx - Mathematics and Statistics

Advertisements
Advertisements
Sum

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

Advertisements

Solution

Let I = `int x^2.cos^-1 x*dx`

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

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

= `(cos^-1x) (x^3/3) - int ((-1)/sqrt(1 - x^2)) (x^3/3)*dx`

= `x^3/(3) cos^-1x + (1)/(3) int (x^2.x)/sqrt(1 - x^2)*dx`

In `int x^3/sqrt(1 - x^2)*dx`, put 1 – x2 = t

∴ – 2x.dx= dt
∴ x.dx = `-(1)/(2)dt`

Also, x2 = 1 – t

∴ I = `x^3/(3) cos^-1x + (1)/(3) int ((1 - t))/sqrt(t) (-1/2)*dt`

= `x^3/(3) cos^-1x - (1)/(6) int (1/sqrt(t) - sqrt(t))*dt`

= `x^3/(3) cos^-1x - (1)/(6) int t^(-1/2) dt + (1)/(6) int t^(1/2)*dt`

= `x^3/(3) cos^-1x - (1)/(6) (t^(1/2)/(1/2)) + (1)/(6) t^(3/2)/(3/2) + c`

= `x^3/(3) cos^-1x - (1)/(3)sqrt(1 - x^2) + (1)/(9)(1 - x^2)^(3/2) + c`.

  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.3 [Page 137]

APPEARS IN

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

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


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 3x.


Integrate the function in `x^2e^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 tan-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 x (log x)2.


Integrate the function in (x2 + 1) log x.


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


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


Integrate the function in `((x- 3)e^x)/(x - 1)^3`.


Integrate the function in e2x sin x.


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`


Find : 

`∫(log x)^2 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 sec^3x.dx`


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


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


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


Evaluate the following : `int x.cos^3x.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 : `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 : `xsqrt(5 - 4x - x^2)`


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 


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)]`


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` =


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` =


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 : `(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 : cot–1 (1 – x + x2)


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


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 : sec4x cosec2x


Solve the following differential equation.

(x2 − yx2 ) dy + (y2 + xy2) dx = 0


Evaluate the following.

`int "x"^2 "e"^"4x"`dx


Evaluate the following.

`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx


Choose the correct alternative from the following.

`int (1 - "x")^(-2) "dx"` = 


Choose the correct alternative from the following.

`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5  "dx"` = 


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


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


Evaluate: `int "dx"/(5 - 16"x"^2)`


Evaluate: ∫ (log x)2 dx


`int (sinx)/(1 + sin x)  "d"x`


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


`int (cos2x)/(sin^2x cos^2x)  "d"x`


`int sin4x cos3x  "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^2 - "a"^2)  "d"x` = ______ + c


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


`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_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.


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


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


`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`


`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`


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 the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.


`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 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 (sin^-1x)/(1 - x^2)^(3//2) dx`.


Evaluate the following.

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


`int(xe^x)/((1+x)^2)  dx` = ______


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


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 (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`


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 the following.

`intx^3  e^(x^2) dx`


Evaluate the following.

`intx^3/sqrt(1+x^4)dx`


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


If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)


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


Share
Notifications



      Forgot password?
Use app×