Advertisements
Advertisements
Question
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Advertisements
Solution
Let I = `int 1/("x"("x"^5 + 1))` dx
∴ I = `int "x"^4/("x"^5("x"^5 + 1))` dx
Put x5 = t
∴ `5"x"^4 "dx" = "dt"`
∴ `"x"^4 "dx" = "dt"/5`
∴ I = `int 1/("t"("t + 1")) * "dt"/5`
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 = `1/5 int (1/"t" + (-1)/"t + 1")` dt
`= 1/5 [int 1/"t" "dt" - int 1/("t + 1") "dt"]`
`= 1/5 [log |"t"| - log |"t" + 1|]` + c
`= 1/5 log |"t"/"t + 1"|` + c
∴ I = `1/5 log |"x"^5/("x"^5 + 1)|` + c
APPEARS IN
RELATED QUESTIONS
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (xdx)/((x - 1)(x - 2))` equals:
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
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:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`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 1/(4x^2 - 20x + 17) "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 "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int ("d"x)/(x^3 - 1)`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
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 = ?
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
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`
What is the result of dividing \[x^{2}+1\] by \[x^{2}-5x+6\]?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
