Advertisements
Advertisements
प्रश्न
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
Advertisements
उत्तर
Let I = `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`= int ("3x" - 1)/(("x - 1")("2x + 1"))` dx
Let `(3"x" - 1)/(("x - 1")("2x" + 1)) = "A"/"x - 1" + "B"/"2x + 1"`
∴ 3x - 1 = A(2x + 1) + B(x - 1) ...(i)
Putting x = 1 in (i), we get
3(1) - 1 = A(2 + 1) + B(0)
∴ 2 = 3A
∴ A = `2/3`
Putting x = `- 1/2` in (i), we get
`3(- 1/2) - 1 = "A"(0) + "B"[- 1/2 - 1]`
∴ `- 5/2 = "B" (- 3/2)`
∴ B = `5/3`
∴ `(3"x" - 1)/(("x" - 1)("2x" + 1)) = (2/3)/("x - 1") + (5/3)/("2x + 1")`
∴ I = `int ((2/3)/("x - 1") + (5/3)/("2x" + 1))` dx
`= 2/3 int 1/("x - 1") "dx" + 5/3 int 1/("2x + 1")`dx
∴ I = `2/3 log |"x - 1"| + 5/3 (log |("2x" + 1)|)/2` + c
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int (2x - 7)/sqrt(4x- 1) dx`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate `int x^2"e"^(4x) "d"x`
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
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
