Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Advertisements
उत्तर
Let I = `int ((x - 1)^2)/(x^2 + 1)^2.dx`
= `int (x^2 - 2x + 1)/(x^2 + 1)^2.dx`
= `int ((x^2 + 1) - 2x)/(x^2 + 1)^2.dx`
= `int [(x^2 + 1)/(x^2 + 1)^2 - (2x)/(x^2 + 1)^2].dx`
= `int (1)/(x^2 + 1)dx - int (2x)/(x^2 + 1)^2.dx`
= I1 – I2 ...(Let)
In I2, Put x2 + 1 = t
∴ 2x dx = dt
= I = `int (1)/(x^2 + 1).dx - int t^-2 dt`
= `tan^-1 x - t^-1/((-1)) + c`
= `tan^-1 x + (1)/(x^2 + 1) + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate `int 1/(x (x - 1))` dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
Evaluate: `int "e"^sqrt"x"` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int x/(x + 2) "d"x`
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int ("d"x)/(x(x^4 + 1))` = ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
`int x^3 e^(x^2) dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
