Advertisements
Advertisements
Question
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Advertisements
Solution
Let I = `int (x^2 + 3)/((x^2 - 1)(x^2 - 2)).dx`
`(x^2 + 3)/((x^2 - 1)(x^2 - 2)) = "A"/(x^2 - 1) + "B"/(x^2 - 2)`
∴ x2 + 3 = A(x2 - 2) + B(x2 - 1)
Put x2 - 1 = 0 i. e. x2 = 1
∴ 1 + 3 = A(1 - 2) + B(1 - 1)
∴ 1 + 3 = A(1 - 2) + 0
∴ 4 = A × -1
∴ A = - 4
Put x2 - 2 = 0 i. e. x2 = 2
∴ `2 + 3 = 0 + B (2 - 1)`
∴ 5 = B × 1
∴ B = 5
I = `int (- 4)/(x^2 - 1^2) "dx" + int 5/(x^2 - (sqrt(2))^2)` dx
I = `- 4 xx 1/(2 xx 1) log |(x - 1)/(x + 1)| + 5 xx 1/(2 xx sqrt2) log |(x - sqrt2)/(x + sqrt2)|` + c ...`[int 1/(x^2 - a^2) dx = 1/(2a) log |(x - a)/(x + a)| + "c"]`
I = `- 2 log |(x - 1)/(x + 1)| + 5/(2 sqrt2) log |(x - sqrt2)/(x + sqrt2)| + c`
APPEARS IN
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Find: `I=intdx/(sinx+sin2x)`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
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:
`1/(e^x -1)`[Hint: Put ex = t]
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 : `(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 : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`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 sin(logx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x sin2x cos5x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
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
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
Evaluate `int x log x "d"x`
`int 1/(4x^2 - 20x + 17) "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/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
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.
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
