Advertisements
Advertisements
Question
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
Advertisements
Solution
Let I = `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
Put log x = t
∴ `1/"x"` dx = dt
∴ I = `int (1 + "t")/((3 + "t")(2 + "3t"))` dt
Let `(1 + "t")/((3 + "t")(2 + "3t")) = "A"/("3 + t") + "B"/(2 + "3t")`
∴ 1 + t = A(2 + 3t) + B(3 + t) ...(i)
Putting t = – 3 in (i), we get
1 -3 = A(2 - 9) + B(0)
∴ - 2 = A (- 7)
∴ A = `2/7`
Putting t = `- 2/3` in (i), we get
`1 - 2/3 = "A"(0) + "B"(3 - 2/3)`
∴ `1/3 = "B"(7/3)`
∴ B = `1/7`
∴ `("1+t")/(("3 + t")("2 + 3t")) = (2/7)/("3 + t") + (1/7)/(2 + "3t")`
∴ I = `int ((2/7)/("3 + t") + (1/7)/("2 + 3t"))` dt
`= 2/7 int 1/(3+"t") "dt" + 1/7 int 1/(2 + "3t")` dt
`= 2/7 log |3 + "t"| + 1/7 * (log |2 + "3t"|)/3` + c
∴ I = `2/7 log |3 + log "x"| + 1/21 log |2 + 3 log "x"| + "c"`
APPEARS IN
RELATED QUESTIONS
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
`int (xdx)/((x - 1)(x - 2))` equals:
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int x^7/(1 + x^4)^2 "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
