Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Advertisements
उत्तर
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
संबंधित प्रश्न
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int 1/(sqrt("x") + "x")` dx
`int x^2/sqrt(1 - x^6)` dx = ________________
`int cos sqrtx` dx = _____________
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int1/(4 + 3cos^2x)dx` = ______
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
