Advertisements
Advertisements
प्रश्न
Integrate the function:
`(x^2 + x + 1)/((x + 1)^2 (x + 2))`
Advertisements
उत्तर
Let `I = (x^2 + x + 1)/((x + 1)^2 (x + 2)) dx`
Now, `(x^2 + x + 1)/((x + 1)^2 (x + 2))`
`= A/(x + 2) + B/(x + 1) + C/(x + 1)^2`
∴ x2 + x + 1 ≡ A(x + 1)2 + B(x + 2)(x + 1) + C(x + 2)
put x = -2.
⇒ 4 - 2 + 1 = A(- 1)2
and 3 = A
A = 3
Put x = - 1
⇒ 1 - 1 + 1 = C(- 1 + 2)
∴ C = 1
Comparing the coefficient of x2,
1 = A + B
B = 1 - A = 1 - 3
∴ B = - 2
∴ `(x^2 + x + 1)/((x + 1)^2 - (x + 2))`
`= 3/(x + 2) - 2/(x + 1) + 1/(x + 1)^2`
∴ I = `int (x^2 + x + 1)/((x + 1)^2 - (x + 2)) dx`
`= int 3/(x + 2) dx - 2int 1/(x + 1) dx + int 1/(x + 1)^2 dx`
`= 3 log (x + 2) - 2 log (x + 1) - 1/(x + 1) + C`
APPEARS IN
संबंधित प्रश्न
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.
Cos 3x
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:
`intx^2 (1 - 1/x^2)dx`
Find the following integrals:
`int (x^3 + 5x^2 -4)/x^2 dx`
Find the following integrals:
`int(1 - x) sqrtx 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:
`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]`
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:
`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:
`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:
`(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
`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 (dx)/(x(x^2 + 1))` equals
`int e^x sec x(1 + tanx) dx` equals
`int sqrt(1 + x^2) dx` is equal to
`d/(dx)x^(logx)` = ______.
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
