Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Advertisements
उत्तर
Let I = `int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Put x = sin θ
⇒ dx = cos θ dθ
I = `int (sin^-1(sin theta))/((1 - sin^2 theta)^(3/2)) * cos theta "d"theta`
= `int (theta * cos theta "d"theta)/((cos^2 theta)^(3/2))`
= `int (theta * cos theta)/(cos^3 theta) "d"theta`
= `int theta/(cos^2 theta) "d"theta`
= `int theta_"I" sec_"II"^2theta "d"theta`
=`theta * sec^2theta "d"theta - int ("D"(theta) * int sec^2theta "d"theta)"d"theta` .....`[because int "u"_"I" * "v"_"II" "d"x = "u" * int "v" "d"x - int ("D"("u") int "v" "dv")"dv" + "C"]`
= `theta * tan theta - int 1 * tan theta "d"theta`
= `theta * tan theta - log sec theta + "C"`
= `sin^-1x * x/sqrt(1 - x^2) - log|sqrt(1 - x^2)| + "C"` ......`[("When" x = sin theta),(therefore tan theta = x/sqrt(1 - x^2) "and" sec theta = sqrt(1 - x^2))]`
Hence, I = `(x sin^-1x)/sqrt(1 - x^2) - log|sqrt(1 - x^2)| + "C"`
APPEARS IN
संबंधित प्रश्न
Integrate the function in x log x.
Integrate the function in x log 2x.
Integrate the function in x (log x)2.
Integrate the function in (x2 + 1) log x.
Integrate the function in ex (sinx + cosx).
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
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)]`
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` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^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
`int 1/(4x + 5x^(-11)) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sin4x cos3x "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Evaluate `int 1/(4x^2 - 1) "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int 1/sqrt(x^2 - a^2)dx` = ______.
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 ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`inte^(xloga).e^x dx` is ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int e^(logcosx)dx`
Evaluate:
`int (logx)^2 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/sqrt(1+x^4)dx`
The value of `inta^x.e^x dx` equals
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
