Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Advertisements
उत्तर
Let I = `int (1)/(x(x^5 + 1)).dx`
= `int x^4/(x^5(x^5 + 1)).dx`
Put x5 = t.
Then 5x4 dx = dt
∴ x4 dx= `dt/(5)`
∴ I = `int (1)/(t(t + 1)).dt/(5)`
= `(1)/(5) int ((t + 1) - t)/(t(t + 1)).dt`
= `(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`
= `(1)/(5)log|x^5/(x^5 + 1)| + c`.
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (xdx)/((x - 1)(x - 2))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^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 : `(1)/(2sinx + sin2x)`
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: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
`int x^7/(1 + x^4)^2 "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int xcos^3x "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 = ?
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
`int x/((x - 1)^2 (x + 2)) "d"x`
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)
Evaluate: `int (dx)/(2 + cos x - sin x)`
`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(5x^2-6x+3)/(2x-3)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
