Advertisements
Advertisements
Question
Integrate the function:
`1/(x - x^3)`
Advertisements
Solution
Let `1/(x - x^3) = 1/(x(1 + x)(1 - x))`
`≡ A/x + B/(1 + x) + C/(1 - x)`
⇒ 1 = A (1 + x) (1 – x) + Bx (1 – x) + Cx (1 + x) …(1)
Putting x = 0 in equation (1),
1 = A(1 + 0) (1 – 0)
⇒ A = 1
Putting x = -1 in equation (1),
1 = B (-1) (1 + 1)
`=> B = - 1/2`
Putting x = 1 in equation (1),
1 = C(1)(1 + 1)
`=> C = 1/2`
`therefore 1/(x - x^3) = 1/x - 1/(2(1 + x)) + 1/(2(1 - x))`
`therefore int 1/(x - x^3) dx = int 1/x dx - 1/2 int 1/(1 + x) dx + 1/2 int 1/(1 - x) dx`
`= log |x| - 1/2 log |1 + x| - 1/2 log |1 - x| + C`
`= 1/2 log |x|^2 - 1/2 log |1 - x^2| + C`
`= 1/2 log |x^2/(1 - x^2)|` + C
APPEARS IN
RELATED QUESTIONS
Evaluate : `∫(sin^6x+cos^6x)/(sin^2x.cos^2x)dx`
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
If `f(x) =∫_0^xt sin t dt` , then write the value of f ' (x).
Find an anti derivative (or integral) of the following function by the method of inspection.
e2x
Find an anti derivative (or integral) of the following function by the method of inspection.
(axe + b)2
Find the following integrals:
`int (4e^(3x) + 1)`
Find the following integrals:
`int (ax^2 + bx + c) dx`
Find the following integrals:
`int (x^3 + 5x^2 -4)/x^2 dx`
Find the following integrals:
`int(2x^2 - 3sinx + 5sqrtx) dx`
Find the following integrals:
`int(sec^2x)/(cosec^2x) dx`
Integrate the function:
`1/(sqrt(x+a) + sqrt(x+b))`
Integrate the function:
`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`
Integrate the function:
`sinx/(sin (x - a))`
Integrate the function:
`cos^3 xe^(log sinx)`
Integrate the function:
f' (ax + b) [f (ax + b)]n
Integrate the functions `(sin^(-1) sqrtx - cos^(-1) sqrtx)/ (sin^(-1) sqrtx + cos^(-1) sqrtx) , x in [0,1]`
Integrate the function:
`(2+ sin 2x)/(1+ cos 2x) e^x`
Integrate the function:
`(sqrt(x^2 +1) [log(x^2 + 1) - 2log x])/x^4`
Evaluate `int(x^3+5x^2 + 4x + 1)/x^2 dx`
Find : \[\int\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}dx\] .
Evaluate: `int (1 - cos x)/(cos x(1 + cos x)) dx`
The anti derivative of `(sqrt(x) + 1/sqrt(x))` is equals:
If `d/(dx) f(x) = 4x^3 - 3/x^4`, such that `f(2) = 0`, then `f(x)` is
`int (dx)/sqrt(9x - 4x^2)` equals
`int (xdx)/((x - 1)(x - 2))` equals
`int sqrt(1 + x^2) dx` is equal to
What is anti derivative of `e^(2x)`
In \[\int f(x)\,dx\], what is \(f(x)\) called?
In \[\int f(x)\,dx\], what is \(x\) called?
What is \(C\) called in \[\int f(x)\,dx=F(x)+C\]?
Which statement about indefinite integrals is correct?
Which antiderivative has derivative \(1\)?
Evaluate \[\int \text{cosec}^2 x\,dx\]
Evaluate \[\int \sec x\tan x\,dx\]
Evaluate \[\int \frac{1}{x}\,dx\]
