Advertisements
Advertisements
प्रश्न
`int tan^-1 sqrt(x) "d"x` is equal to ______.
पर्याय
`(x + 1) tan^-1 sqrt(x) - sqrt(x) + "C"`
`x tan^-1 sqrt(x) - sqrt(x) + "C"`
`sqrt(x) - x tan^-1 sqrt(x) + "C"`
`sqrt(x) - (x + 1) tan^-1 sqrt(x) + "C"`
Advertisements
उत्तर
`int tan^-1 sqrt(x) "d"x` is equal to `(x + 1) tan^-1 sqrt(x) - sqrt(x) + "C"`.
Explanation:
Let I = `int 1 * tan^-1 sqrt(x) "d"x`
= `tan^-1 sqrt(x) int 1 "d"x - int[(tan^-1 sqrt(x))"'" int 1"d"x]"d"x`
= `tan^-1 sqrt(x) * x - int 1/(1 + x) * 1/(2sqrt(x)) * x"d"x` ....[Integrating by parrts]
= `xtan^-1 sqrt(x) - 1/2 int sqrt(x)/(1 + x) "d"x`
Put x = t2
⇒ dx = 2t dt
∴ I = `xtan^-1 sqrt(x) - int "t"^2/(1 + "t"^2) "d"x`
= `xtan^-1 sqrt(x) - int (1 - 1/(1 + "t"^2))"dt"`
= `xtan^-1 sqrt(x) - "t" + tan^-1 1 + "C"`
= `xtan^-1 sqrt(x) - sqrt(x) + tan^-1 sqrt(x) + "C"`
= `(x + 1) tan^-1 sqrt(x) - sqrt(x) + "C"`
APPEARS IN
संबंधित प्रश्न
Integrate the function in x 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 sin 3x dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following: `int logx/x.dx`
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 : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
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 (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
`int (sinx)/(1 + sin x) "d"x`
`int 1/(4x + 5x^(-11)) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Evaluate `int 1/(x(x - 1)) "d"x`
`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`
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
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 ______.
`int(1-x)^-2 dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate:
`int1/(x^2 + 25)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
The value of `inta^x.e^x dx` equals
