Advertisements
Advertisements
प्रश्न
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Advertisements
उत्तर
Let `I = int (4x + 2) sqrt(x^2 + x + 1)` dx
or `I = 2 int (2x + 1) sqrt ((x^2 + x + 1))` dx
Taking x2 + x + 1 = t
2x + 1 = dt
Hence, `I = 2 int sqrt t dt`
`= 2 int t^(1/2) dt = 2. 2/3 t^(3/2) + C`
`= 4/3 (x^2 + x + 1)^(3/2) + C`
APPEARS IN
संबंधित प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`(1+ log x)^2/x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int "e"^sqrt"x"` dx
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
`int (logx)^2/x dx` = ______.
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
