Advertisements
Advertisements
Question
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Solution
Let I = `int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Put log x = t
∴ `1/x "d"x = dt`
∴ I = `int (2"t" + 3)/((3"t" + 2)("t"^2 + 1)) "dt"`
Let `(2 + 3)/((3"t" + 2)("t"^2 + 1)) = "A"/(3"t" + 2) + ("Bt" + "C")/("t"^2 + 1)`
∴ 2t + 3 = A(t2 + 1) + (Bt + C)(3t + 2) .........(i)
Putting t = `-2/3` in (i), we get
`2((-2)/3) + 3 = "A"[((-2)/3)^2 + 1]`
∴ `(-4)/3 + 3 = "A"(4/9 + 1)`
∴ `5/3 = "A"(13/9)`
∴ A = `15/13`
Putting t = 0 in (i), we get
3 = A(1) + C(2)
∴ 3 = `15/13 + 2"C"`
∴ `3 - 15/13` = 2C
∴ `24/13` = 2C
∴ C = `12/13`
Putting t = 1 in (i), we get
2 + 3 = A(1 + 1) + (B + C)(3 + 2)
∴ 5 = 2A + 5(B + C)
∴ 5 = `2(15/13) + 5("B" + 12/13)`
∴ 5 = `30/13 + 5"B" + 60/13`
∴ 5B = `5 - 30/13 - 60/13`
∴ 5B = `-25/13`
∴ B = `(-5)/13`
∴ `(2"t" + 3)/((3"t" + 2)("t"^2 + 1)) = (15/13)/(3"t" + 2) + (-5/13 "t" + 12/13)/("t"^2 + 1)`
∴ I = `int((15/13)/(3"t" + 2) + ((-5)/13 "t" + 12/13)/("t"^2 + 1)) "dt"`
= `15/13 int 1/(3"t" + 2) "dt" - 5/13 int "t"/("t"^2 + 1) "dt" + 12/13 int 1/("t"^2 + 1) "dt"`
= `15/13 int 1/(3"t" + 2) "dt" - 5/13*1/2 int (2"t")/("t"^2 + 1) "dt" + 12/13 int 1/("t"^2 + 1) "dt"`
= `15/13* (log|3"t" + 2|)/3 - 5/26 log|"t"^2 + 1| + 12/13 tan^-1 "t" + "c"`
∴ I = `5/13 log |3 log x 2| - 5/26 log |(logx)^2 + 1| + 12/13 tan^-1(logx) + "c"`
RELATED QUESTIONS
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Integrate the following w.r.t.x : `(x + 5)/(x^3 + 3x^2 - x - 3)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int 1/(x(x^3 - 1)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sec^3x "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int 1/(sinx(3 + 2cosx)) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
`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 x^2"e"^(4x) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
Evaluate: `int (dx)/(2 + cos x - sin x)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
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(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
