Advertisements
Advertisements
Question
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Advertisements
Solution
Let I = `int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Resolving into partial fraction, we put
`(2x - 1)/((x - 1)(x + 2)(x - 3)) = "A"/(x - 1) + "B"/(x + 2) + "C"/(x - 3)`
⇒ 2x – 1 = A(x + 2)(x – 3) + B(x – 1)(x – 3) + C(x – 1)(x + 2)
Put x = 1
1 = A(3)(– 2)
⇒ A = `-1/6`
Put x = – 2
– 5 = B(– 3)(– 5)
⇒ B = `- 1/3`
Put x = 3
5 = C(2)(5)
⇒ C = `1/2`
∴ `int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x = - 1/6 int 1/(x - 1) "d"x - 1/3 int 1/(x + 2) "d"x + 1/2 int 1/(x - 3) "d"x`
= `- 1/6 log |x - 1| - 1/3 log|x + 2| + 1/2 log|x - 3| + "C"`
= `- log|x - 1|^(1/6) - log(x + 2)^(1/3) + log(x - 3)^(1/3) + "C"`
Hence, `int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x = log[sqrt(x - 3)/((x - 1)^(1/6) (x + 2)^(1/3))] + "C"`
APPEARS IN
RELATED QUESTIONS
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
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:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (dx)/(x(x^2 + 1))` equals:
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int sqrt(4^x(4^x + 4)) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int x^3tan^(-1)x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "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 = ?
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
Evaluate `int x^2"e"^(4x) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
Evaluate: `int (dx)/(2 + cos x - sin x)`
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 = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
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
A proper rational function can be expressed as a sum of simpler rational functions called what?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
What is the result of dividing \[x^{2}+1\] by \[x^{2}-5x+6\]?
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
What must be included for a repeated linear factor?
