Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Advertisements
उत्तर
Let I = `int "x"^5/("x"^2 + 1)`dx
`int (("x"^2)^2 * "x")/("x"^2 + 1)`dx
Put x2 + 1 = t
∴ 2x . dx = dt
∴ x . dx = `1/2 * "dt"`
Also, x2 = t - 1
∴ I = `int ("t" - 1)^2/"t" * 1/2`dt
`= 1/2 int ("t"^2 - 2"t" + 1)/"t"`dt
`= 1/2 int ("t" - 2 + 1/"t")`dt
`= 1/2 ["t"^2/2 - 2"t" + log |"t"|]` + c
`= 1/4 "t"^2 - "t" + 1/2 log |"t"| + "c"`
∴ I = `1/4 ("x"^2 + 1)^2 - ("x"^2 + 1) + 1/2 log |"x"^2 + 1|` + c
Notes
The answer in the textbook is incorrect.
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`cos sqrt(x)/sqrtx`
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Integrate the following w.r.t. x : x3 + x2 – x + 1
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Evaluate: `int log ("x"^2 + "x")` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int logx/x "d"x`
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int sin^2(x/2)dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate:
`int sin^3x cos^3x dx`
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int 1/(x(x-1))dx`
Evaluate `int 1/(x(x-1)) dx`
