Advertisements
Advertisements
Question
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Advertisements
Solution
Let I = `int (sin^-1x)/(1 - x^2)^(3//2) dx`
Consider t = sin–1 x
`dt/dx = 1/sqrt(1 - x^2)`
∴ I = `int (t.dt)/((1 - x^2))`
= `int (t.dt)/((1 - sin^2t))`
= `int (t.dt)/(cos^2t)`
= `int t . sec^2 t dt`
On integrating by parts
= `t int sec^2t.dt - int {(d(t))/dt int sec^2 t}dt`
= `t tan t - int 1.tan t dt`
= t tan t – log sec t + C
= sin–1x tan [sin–1x] – log sec [sin–1x] + C
APPEARS IN
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`
Integrate the function in x sin x.
Integrate the function in x log 2x.
Integrate the function in x tan-1 x.
Integrate the function in x cos-1 x.
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
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) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int ("d"x)/(x - x^2)` = ______
`int"e"^(4x - 3) "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 cot "x".log [log (sin "x")] "dx"` = ____________.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int 1/sqrt(x^2 - a^2)dx` = ______.
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 ______.
`int(1-x)^-2 dx` = ______
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int (logx)^2 dx`
Evaluate `int tan^-1x dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x sqrt(1 + x^2) dx`
The value of `inta^x.e^x dx` equals
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
