Advertisements
Advertisements
Question
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.
Options
True
False
Advertisements
Solution
True
Explanation:
Let I =`(("x - 1"))/(("x + 1")("x - 2"))` dx
Let `(("x - 1"))/(("x + 1")("x - 2")) = "A"/"x + 1" + "B"/"x - 2"`
∴ x - 1 = A(x - 2) + B(x + 1) ....(i)
Putting x = –1 in (i), we get
- 1 - 1 = A(- 1 - 2)
∴ - 2 = - 3A
∴ A = `2/3`
Putting x = 2 in (i), we get
2 - 1 = B(2 + 1)
∴ 1 = 3B
∴ B = `1/3`
∴ I = `int ((2/3)/("x + 1") + (1/3)/("x - 2"))` dx
`= 2/3 int 1/"x + 1" "dx" + 1/3 int 1/"x - 2"` dx
`= 2/3 log |"x + 1"| + 1/3 log |"x - 2"|` + c
Comparing the above with
A log |x + 1| + B log |x - 2| + c, we get
∴ A = `2/3, "B" = 1/3`
∴ A + B = `2/3 + 1/3 = 1`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
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 : `x^2/sqrt(1 - x^6)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int (sinx)/(sin3x) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int 1/(sinx(3 + 2cosx)) "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 sqrt(1 + x) "d"x` =
`int x/((x - 1)^2 (x + 2)) "d"x`
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`int(5x^2-6x+3)/(2x-3)dx`
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
