Advertisements
Advertisements
प्रश्न
Integrate the function:
`1/(x^(1/2) + x^(1/3)) ["Hint:" 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1+x^(1/6))), "put x" = t^6]`
Advertisements
उत्तर
Let I = `int 1/(x^(1/2) + x^(1/3))`dx
Put `x^(1/6) = t, x = t^6 `
dx = 6 t5 dt
`int = (6 t^5 dt)/(t^3 + t^2) = 6 int (t^3/(t + 1))`dt
`= 6 int (t^3 + 1 - 1)/(t + 1)`dt
`= 6 int ((t^3 + 1)/(t + 1) - 1/(t + 1)) dt`
`= int (((t + 1)(t^2 - t - 1))/(t + 1) - 1/(t + 1)) dt`
`= 6 int (t^2 - t + 1 - 1/(t + 1)) dt`
`= 6 int t^2 dt - 6 int t dt - 6 int 1/(t + 1) dt`
`= (6t^3)/3 - (6t^2)/2 - 6 log |t + 1| + C`
= 2t3 - 3t2 - 6 log |t + 1| + C
= `2sqrtx - 3x^(1/3) + 6x^(1/6) - 6 log |x^(1/6) + 1| + C`
APPEARS IN
संबंधित प्रश्न
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
Find an anti derivative (or integral) of the following function by the method of inspection.
sin 2x
Find an anti derivative (or integral) of the following function by the method of inspection.
e2x
Find an antiderivative (or integral) of the following function by the method of inspection.
sin 2x – 4 e3x
Find the following integrals:
`int (4e^(3x) + 1)`
Find the following integrals:
`intx^2 (1 - 1/x^2)dx`
Find the following integrals:
`int (ax^2 + bx + c) dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Find the following integrals:
`int(1 - x) sqrtx dx`
Find the following integrals:
`int(2x^2 - 3sinx + 5sqrtx) dx`
Find the following integrals:
`intsec x (sec x + tan x) dx`
Find the following integrals:
`int (2 - 3 sinx)/(cos^2 x) dx.`
If `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0, then f(x) is ______.
Integrate the function:
`1/(x - x^3)`
Integrate the function:
`sinx/(sin (x - a))`
Integrate the function:
`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`
Integrate the function:
`1/(cos (x+a) cos(x+b))`
Integrate the function:
`x^3/(sqrt(1-x^8)`
Integrate the function:
`cos^3 xe^(log sinx)`
Integrate the function:
`(2+ sin 2x)/(1+ cos 2x) e^x`
Integrate the function:
`(x^2 + x + 1)/((x + 1)^2 (x + 2))`
If `d/(dx) f(x) = 4x^3 - 3/x^4`, such that `f(2) = 0`, then `f(x)` is
`sqrt((10x^9 + 10^x log e^10)/(x^10 + 10^x)) dx` equals
`int (dx)/(sin^2x cos^2x) dx` equals
`int (sin^2x - cos^2x)/(sin^2x cos^2x) dx` is equal to
`int (e^x (1 + x))/(cos^2 (xe^x)) dx` equal
`int (dx)/sqrt(9x - 4x^2)` equal
`int (xdx)/((x - 1)(x - 2))` equals
`int sqrt(x^2 - 8x + 7) dx` is equal to:-
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
What is \(C\) called in \[\int f(x)\,dx=F(x)+C\]?
Evaluate \[\int \sin x\,dx\]
Evaluate \[\int \text{cosec}^2 x\,dx\]
Which expression is also an antiderivative of \[\frac{1}{\sqrt{1-x^2}}\]?
