Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Advertisements
उत्तर
Let I = `int (2x)/(4 - 3x - x^2).dx`
Let `(2x)/(4 - 3x - x^2)`
= `(2x)/((4 + x)(1 - x)`
= `"A"/(4 + x) + "B"/(1 - x)`
∴ 2x = A(1 – x) + B(4 + x)
Put 4 + x = 0, i.e. x = – 4, we get
– 8 = A(5) + B(0)
∴ A = `-(8)/(5)`
Put 1 – x = 0, i.e x = 1, we
2 = A(0) + B(5)
∴ B = `(2)/(5)`
∴ `(2x)/(4 - 3x - x^2) = ((-8/5))/(4 + x) + ((2/5))/(1 - x)`
∴ I = `int [((-8)/5)/(4 + x) + ((2/5))/(1 - x)].dx`
= `-(8)/(5) int(1)/(4 + x).dx + (2)/(5) int (1)/(1 - x).dx`
= `-(8)/(5)log|4 + x| + (2)/(5).(log|1 - x|)/(-1) + c`
= `-(8)/(5)log|4 + x| - (2)/(5)log|1 - x| + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
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)/(sin2x + cosx)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
`int (2x - 7)/sqrt(4x- 1) dx`
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x sin2x cos5x "d"x`
`int ("d"x)/(x^3 - 1)`
`int xcos^3x "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "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`
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 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
`int 1/(x^2 + 1)^2 dx` = ______.
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
