Advertisements
Advertisements
प्रश्न
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
उत्तर
∴ `2/((1 - x)(1 + x^2)) = "A"/(1 - x) + ("B"x + "C")/(1 + x^2)` ...(1)
`\implies` 2 = A(1 + x2) + (Bx + C)(1 – x)
`\implies` 2 = A(1 + x2) + (– Bx2 + Bx – Cx + C)
`\implies` 2 = (A – B)x2 + (B – C)x + A + C
Equating the coefficients of like terms.
A – B = 0,
B – C = 0
and A + C = 2
On adding,
2A = 2
`\implies` A = 1
∴ B = A = 1
and C = B = 1
By (1) `2/((1 - x)(1 + x^2)) = 1/(1 - x) + (x + 1)/(1 + x^2)`
Now integrating
`int 2/((1 - x)(1 + x^2))"d"x = int 1/(1 - x)"d"x + int (x + 1)/(1 + x^2)"d"x`
= `int 1/(1 - x)"d"x + int x/(1 + x^2)"d"x + int 1/(1 + x^2)"d"x`
= `-log(1 - x) + 1/2log(1 + x^2) + tan^-1x + "C"`.
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int 1/(x(x^3 - 1)) "d"x`
`int sec^3x "d"x`
`int sin(logx) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int x^2"e"^(4x) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
