Advertisements
Advertisements
Question
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Advertisements
Solution
Let I = `int 1/(x(x^6 + 1))` dx
`= int x^5/(x^6(x^6 + 1))`dx
Put x6 = t
∴ 6x5 dx = dt
∴ `x^5 * dx = 1/6 * dt`
∴ I = `1/6 int dt/(t(t + 1))`
`= 1/6 int ((t + 1) - t)/(t(t + 1))` dt
`= 1/6 int (1/t - 1/(t + 1))` dt
= `1/6` [log | t | - log |t + 1|] + c
`= 1/6 log |t/(t + 1)|` + c
∴ I = `1/6 log |x^6/(x^6 + 1)|` + c
RELATED QUESTIONS
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int cos sqrtx` dx = _____________
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int(1 - x)^(-2) dx` = ______.
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate `int (1)/(x(x - 1))dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
