Advertisements
Advertisements
प्रश्न
Integrate the function:
`1/((x^2 + 1)(x^2 + 4))`
Advertisements
उत्तर
Let `I = 1/((x^2 + 1)(x^2 + 4)) = 1/((y + 1)(y + 4))`
Where x2 = y
`= A/(y + 1) + B/(y + 4)`
`=> A(y + 4) + B(y + 1)` ...(1)
Putting y = -1 in equation (1),
∴ 1 = A(- 1 + 4)
`=> A = 1/3`
Putting y = -4 in equation (1),
∴ 1 = B(- 4 + 1)
`=> B = - 1/3`
`therefore 1/((x^2 + 1)(x^2 + 4)) = 1/(3 (y + 1)) - 1/(3 (y + 4))`
`= 1/(3 (x^2 + 1)) - 1/(3 (x^2 + 4))`
Now, `I = int [1/ (3(x^2 + 1)) - 1/ (3(x^2 + 4))] dx`
`= (1/3 tan^-1 x) - (1/3 xx 1/2 tan^-1 (x/2)) + C`
`= 1/3 tan^-1 x - 1/6 tan^-1 (x/2) + C`
APPEARS IN
संबंधित प्रश्न
Write the antiderivative of `(3sqrtx+1/sqrtx).`
Evaluate : `∫(sin^6x+cos^6x)/(sin^2x.cos^2x)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.
(axe + b)2
Find the following integrals:
`intx^2 (1 - 1/x^2)dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Find the following integrals:
`int (x^3 - x^2 + x - 1)/(x - 1) dx`
Find the following integrals:
`int(1 - x) sqrtx dx`
Find the following integrals:
`intsqrtx( 3x^2 + 2x + 3) dx`
Find the following integrals:
`intsec x (sec x + tan x) dx`
Find the following integrals:
`int(sec^2x)/(cosec^2x) dx`
Integrate the function:
`(5x)/((x+1)(x^2 +9))`
Integrate the function:
`(e^(5log x) - e^(4log x))/(e^(3log x) - e^(2log x))`
Integrate the function:
`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`
Integrate the function:
`x^3/(sqrt(1-x^8)`
Integrate the function:
`e^x/((1+e^x)(2+e^x))`
Integrate the function:
f' (ax + b) [f (ax + b)]n
Integrate the function:
`1/sqrt(sin^3 x sin(x + alpha))`
Integrate the function:
`sqrt((1-sqrtx)/(1+sqrtx))`
Integrate the function:
`(2+ sin 2x)/(1+ cos 2x) e^x`
Integrate the function:
`tan^(-1) sqrt((1-x)/(1+x))`
Evaluate `int(x^3+5x^2 + 4x + 1)/x^2 dx`
Evaluate `int tan^(-1) sqrtx dx`
Evaluate: `int (1 - cos x)/(cos x(1 + cos x)) dx`
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 (e^x (1 + x))/(cos^2 (xe^x)) dx` equal
`int (xdx)/((x - 1)(x - 2))` equals
`int (dx)/(x(x^2 + 1))` equals
`int x^2 e^(x^3) dx` equals
`int sqrt(1 + x^2) dx` is equal to
`int sqrt(x^2 - 8x + 7) dx` is equal to:-
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
