Evaluate: ∫1x(xn+1) dx - Mathematics and Statistics

Advertisements
Advertisements
Sum

Evaluate: `int 1/("x"("x"^"n" + 1))` dx

Advertisements

Solution

Let I = `int 1/("x"("x"^"n" + 1))` dx

∴ I = `int "x"^"n - 1"/("x"^"n - 1"  xx  "x"("x"^"n" + 1))` dx

∴ I = `int "x"^"n - 1"/("x"^"n" ("x"^"n" + 1))` dx

Put xn = t

∴ `"n""x"^"n - 1"  "dx" = "dt"`

∴ `"x"^"n - 1"  "dx" = "dt"/"n"`

∴ I = `int 1/("t"("t + 1")) * "dt"/"n"`

Let `1/("t"("t + 1")) = "A"/"t" + "B"/"t + 1"`

∴ 1 = A(t + 1) + Bt     ....(i)

Putting t = –1 in (i), we get

1 = A(0) + B(- 1)

∴ 1 = - B

∴ B = - 1

Putting t = 0 in (i), we get

1 = A(1) + B(0)

∴ A = 1

∴ `1/("t"("t + 1")) = 1/"t" + (- 1)/"t + 1"`

∴ I = `1/"n" int (1/"t" + (-1)/"t + 1")` dt

`= 1/"n" [int 1/"t" "dt" - int 1/("t + 1") "dt"]`

`= 1/"n" [log |"t"| - log |"t" + 1|]` + c

`= 1/"n" log |"t"/"t + 1"|` + c

∴ I = `1/"n" log |"x"^"n"/("x"^"n" + 1)|` + c

  Is there an error in this question or solution?
Chapter 5: Integration - EXERCISE 5.6 [Page 135]

RELATED QUESTIONS

Find : `int x^2/(x^4+x^2-2) dx`


Find: `I=intdx/(sinx+sin2x)`


Evaluate: `∫8/((x+2)(x^2+4))dx` 


Integrate the rational functions `x/((x +  1)(x+ 2))`


Integrate the rational functions `1/(x^2 - 9)`


Integrate the rational functions `(3x - 1)/((x - 1)(x - 2)(x - 3))`


Integrate the rational functions `(1 - x^2)/(x(1-2x))`


Integrate the rational functions `x/((x -1)^2 (x+ 2))`


Integrate the rational functions `(3x + 5)/(x^3 - x^2 - x + 1)`


Integrate the rational functions `(x^2 + x + 1)/(x^2 -1)`


Integrate the rational functions `1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]


Integrate the rational functions `((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`


Integrate the rational functions  `(2x)/((x^2 + 1)(x^2 + 3))`


Integrate the rational functions `1/(x(x^4 - 1))`


Choose the correct answer `int (xdx)/((x - 1)(x - 2))` equals

A. `log |(x - 1)^2/(x -2)| + C`

B. `log |(x-2)^2/(x -1)| +C`

C. `log|((x- 1)/(x- 2))^2| + C`

D. log|(x - 1)(x - 2)| + C


Choose the correct answer `int (dx)/(x(x^2 + 1))` equal

A. `log |x| - 1/2 log (x^2 + 1) + C`

B. `log |x| + 1/2 log(x^2 + 1) + C`

C. `- log|x| + 1/2 log (x^2 + 1) + C`

D. `1/2 log|x| + log(x^2 + 1)+ C`


Evaluate : `∫(x+1)/((x+2)(x+3))dx`


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 : `(12x + 3)/(6x^2 + 13x - 63)`


Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`


Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`


Integrate the following w.r.t. x : `(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`


Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`


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 : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`


Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`


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 w.r.t. x : `(1)/(sin2x + cosx)`


Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`


Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`


Choose the correct options from the given alternatives :

If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =


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 : `(cos 7x - cos8x)/(1 + 2 cos 5x)`


Integrate the following w.r.t.x : `x^2/((x - 1)(3x - 1)(3x - 2)`


Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`


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 (2"x" + 1)/(("x + 1")("x - 2"))` dx


Evaluate: `int 1/("x"("x"^5 + 1))` dx


Choose the correct alternative from the following.

`int "dx"/(("x - 8")("x + 7"))`= 


Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx


`int ((2x - 7))/sqrt(4x- 1)  "d"x`


`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`


`int x^7/(1 + x^4)^2  "d"x`


`int x^2sqrt("a"^2 - x^6)  "d"x`


`int sqrt(4^x(4^x + 4))  "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 sqrt((9 + x)/(9 - x))  "d"x`


`int 1/(4x^2 - 20x + 17)  "d"x`


`int sec^2x  "d"x`


`int sec^2x sqrt(tan^2x + tanx - 7)  "d"x`


`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`


`int (x^2 + x -1)/(x^2 + x - 6)  "d"x`


`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)  "d"x`


`int (3x + 4)/sqrt(2x^2 + 2x + 1)  "d"x`


`int x sin2x cos5x  "d"x`


`int (x + sinx)/(1 - cosx)  "d"x`


`int ("d"x)/(x^3 - 1)`


`int (5"e"^x)/(("e"^x + 1)("e"^(2x) + 9))  "d"x`


`int (sin2x)/(3sin^4x - 4sin^2x + 1)  "d"x`


`int  ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1])  "d"x`


Choose the correct alternative:

`int sqrt(1 + x)  "d"x` =


Choose the correct alternative:

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?


`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c


If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + 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 (2"e"^x + 5)/(2"e"^x + 1)  "d"x`


Evaluate `int x log x  "d"x`


`int 1/(4x^2 - 20x + 17)  "d"x`


If `int(sin2x)/(sin5x  sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______


If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______


Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x2 = t


Evaluate the following:

`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`


Evaluate the following:

`int_"0"^pi  (x"d"x)/(1 + sin 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 ______.


The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.


Evaluate: `int (dx)/(2 + cos x - sin x)`


Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)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)


Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.


`int 1/(x^2 + 1)^2 dx` = ______.


If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.


If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1  x/2 + B tan^-1(x/3) + 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 (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`


Find: `int x^4/((x - 1)(x^2 + 1))dx`.


Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.


Evaluate`int(5x^2-6x+3)/(2x-3)dx`


Evaluate: 

`int 2/((1 - x)(1 + x^2))dx`


Evaluate:

`int x/((x + 2)(x - 1)^2)dx`


Evaluate.

`int (5x^2 - 6x + 3) / (2x -3) dx`


Share
Notifications



      Forgot password?
Use app×