Advertisements
Advertisements
Question
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
Advertisements
Solution
Let I = `int (x^2 + x -1)/(x^2 + x - 6) "d"x`
= `int (x^2 + x - 6 + 5)/(x^2 + x - 6) "d"x`
= `int [1 + (5/(x^2 + x - 6))] "d"x`
Let `5/(x^2 + x - 6) = 5/((x + 3)(x - 2))`
= `"A"/(x + 3) + "B"/(x - 2)`
∴ 5 = A(x − 2) + B(x + 3) ........(i)
Putting x = 2 in (i), we get
5 = B(5)
∴ B = 1
Putting x = −3 in (i), we get
5 = A(− 5)
∴ A = −1
∴ `5/((x + 3)(x - 2)) = (-1)/(x + 3) + 1/(x - 2)`
∴ I = `int[1 + (-1)/(x + 3) + 1/(x - 2)] "d"x`
= `int "d"x - int 1/(x + 3) "d"x + int 1/(x - 2) "d"x`
= x − log|x + 3| + log|x − 2| + c
∴ I = `x + log |(x - 2)/(x + 3)| + "c"`
RELATED QUESTIONS
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`2/((1-x)(1+x^2))`
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))`
`int (xdx)/((x - 1)(x - 2))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
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 : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int (2x - 7)/sqrt(4x- 1) dx`
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
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
`int 1/(4x^2 - 20x + 17) "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "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.
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)
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 dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Which expression defines a rational function?
What is the result of dividing \[x^{2}+1\] by \[x^{2}-5x+6\]?
Which factorisation is correct for \[x^{2}-5x+6\]?
What are the values of \[\mathrm{A}\] and \[\mathrm{B}\] in \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\]?
What numerator is used for an irreducible quadratic factor?
