Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`(3x -1)/(x + 2)^2`
उत्तर
Let `I = int (3x - 1)/ (x + 2)^2 dx`
Let `(3x - 1)/(x + 2)^2 = A/(x + 2) + B/ (x + 2)^2`
⇒ 3x - 1 = A (x + 2) + B .....(i)
Comparing coefficients of x in (i), we get
A = 3
Comparing the coefficients of constant terms in (i), we get
2A + B = -1
Put A = 3 in (ii), and we get 6 + B = -1
⇒ B = -7
∴ `(3x - 1)/(x + 2)^2 = 3/ (x + 2) + (-7)/(x + 2)^2`
⇒ `int (3x - 1)/(x + 2)^2 dx = 3 int dx/ (x + 2) - 7 int dx/ (x + 2)^2`
`= 3 log |x + 2| -7 (x + 2)^-1/-1 + C`
`= 3 log |x + 2| + 7/ (x + 2) + C`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(x(x^4 - 1))`
`int (xdx)/((x - 1)(x - 2))` equals:
`int (dx)/(x(x^2 + 1))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Integrate the following w.r.t.x : `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int "x"/(("x - 1")^2("x + 2"))` dx
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate `int x log x "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
