Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Advertisements
उत्तर
Let I = `int (x^2 + x - 1)/(x^2 + x - 6).dx`
= `int ((x^2 + x - 6) + 5)/(x^2 + x - 6).dx`
= `int [1 + (5)/(x^2 + x - 6)].dx`
= `int 1 dx + 5 int (1)/(x^2 + x - 6).dx`
Let `(1)/(x^2 + x - 6)`
= `(1)/((x + 3)(x - 2)`
= `"A"/(x + 3) + "B"/(x- 2)`
∴ 1 = A(x – 2) + B(x + 3)
Put x 3 = 0, i.e. x = –3, we get
1 = A(– 5) + B(0)
∴ A = `(-1)/(5)`
Put x – 2 = 0, i.e. x = 2, we get
1 = A(0) + B(5)
∴ B = `(1)/(5)`
∴ `(1)/(x^2 + x - 6) = ((-1/5))/(x + 3) + ((1/5))/(x - 2)`
∴ I = `int 1 dx + 5 int [((-1/5))/(x + 3) + ((1/5))/(x - 2)].dx`
= `int 1 dx - int (1)/(x + 3).dx + int (1)/(x - 2).dx`
= x – log|x + 3| + log|x – 2| + c
= `x + log|(x - 2)/(x + 3)| + c`.
APPEARS IN
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
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:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
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
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int 1/(4x^2 - 20x + 17) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int xcos^3x "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int x/((x - 1)^2 (x + 2)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
When is a rational function called improper?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
