Advertisements
Advertisements
Question
Integrate the function in x cos-1 x.
Advertisements
Solution
Let `I = int x cos^-1 x dx = int cos^-1 x*x dx`
`= cos^-1 x* int x dx - int [d/dx (cos^-1 x) int x dx] dx`
`= cos^-1 x (x^2/2) - int (-1)/ sqrt (1 - x^2) (x^2/2) dx`
`= x^2/2 cos^-1 x + 1/2 int x^2/ sqrt (1 - x^2) dx`
∴ `I = x^2/2 cos^-1 x+ 1/2 I_1` ....(i)
Where `I_1 = int x^2/ sqrt (1 - x^2) dx`
Put x = cos θ
⇒ dx = -sinθ dθ
∴ `I_1 = int (cos^2 theta (-sin theta))/sqrt (1 - cos^2 theta) d theta`
`= - int cos^2 theta d theta = - 1/2 int (1 + cos 2 theta) d theta`
`= -1/2 (theta + (sin 2 theta)/2) + C`
`= -1/2 (theta + 1/2 xx 2 sin theta cos theta) + C`
`= - 1/2 (theta + cos theta sqrt (1 - cos^2 theta)) + C`
`= - 1/2 (cos^-1 x + x sqrt (1 - x^2)) + C` ....(ii)
From (i) and (ii), we get
`I = (2x^2 - 1) (cos^-1 x)/4 - x/4 sqrt (1 - x^2) + C`
APPEARS IN
RELATED QUESTIONS
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`
Integrate the function in x log x.
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following: `int x.sin^-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`
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 : `xsqrt(5 - 4x - x^2)`
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` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Evaluate the following.
`int "x"^2 *"e"^"3x"`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int 1/(4x^2 - 1) "d"x`
`int cot "x".log [log (sin "x")] "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"` = ______.
Find `int_0^1 x(tan^-1x) "d"x`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
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 ______.
Solution of the equation `xdy/dx=y log y` is ______
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int e^(ax)*cos(bx + c)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)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3e^(x^2) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
