Advertisements
Advertisements
Question
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`.
APPEARS IN
RELATED QUESTIONS
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
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 x2 log x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
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 x.cos^3x.dx`
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 : [2 + cot x – cosec2x]ex
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 with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (sinx)/(1 + sin x) "d"x`
`int sin4x cos3x "d"x`
`int 1/(x^2 - "a"^2) "d"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
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int 1/sqrt(x^2 - a^2)dx` = ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
`int(1-x)^-2 dx` = ______
`int1/sqrt(x^2 - a^2) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`inte^x sinx 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`
Evaluate:
`int1/(x^2 + 25)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).
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
