Advertisements
Advertisements
Question
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Advertisements
Solution
Let I = `int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Put x2 = t for the purpose of partial fraction.
We get `"t"/(("t" + "a"^2)("t" + "b"^2))`
Put `"t"/(("t" + "a"^2)("t" + "b"^2)) = "A"/("T" + "a"^2) + "B"/("t" + "b"^2)`
⇒ `"t"/(("t" + "a"^2)("t" + "b"^2)) = ("A"("t" + "b"^2) + "B"("t" + "a"^2))/(("t" + "a"^2)("t" + "b"^2))`
⇒ t = At + Ab2 + Bt + Ba2
Comparing the like terms, we get
A + B = 1 and Ab2 + Ba2 = 0
A = `(-"a"^2)/"b"^2 "B"`
∴ `(-"a"^2)/"b"^2 "B" + "B"` = 1
`"B"((-"a"^2)/"b"^2 + 1)` = 1
⇒ `"B"((-"a"^2 + "b"^2)/"b"^2)` = 1
⇒ B = `"b"^2/("b"^2 - "a"^2)` and A = `(-"a"^2)/"b"^2 xx "b"^2/("b"^2 - "a"^2) = "a"^2/("a"^2 - "b"^2)`
So A = `"a"^2/("a"^2 - "b"^2)` and B = `(-"b"^2)/("a"^2 - "b"^2)`
∴ `int x^2/((x^2 + "a"^2)(x^2 + "b"^2)) "d"x = "a"^2/("a"^2 - "b"^2) int 1/(x^2 + "a"^2) "d"x - "b"^2/("a"^2 - "b"^2) int 1/(x^2 + "b"^2) "d"x`
= `"a"^2/("a"^2 - "b"^2) xx 1/"a" tan^-1 x/"a" - "b"^2/("a"^2 - "b"^2) * 1/"b" tan^-1 x/"b"`
= `"a"/("a"^2 - "b"^2) tan^-1 x/"a" - "b"/("a"^2 - "b"^2) tan^-1 x-"b" + "C"`
Hence, I = `1/("a"^2 - "b"^2) ["a" tan^-1 x/"a" - "b" tan^-1 x/"b"] + "C"`.
APPEARS IN
RELATED QUESTIONS
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:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
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:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (dx)/(x(x^2 + 1))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int 1/("x"("x"^"n" + 1))` 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
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int x sin2x cos5x "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int xcos^3x "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
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)
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
