Advertisements
Advertisements
Question
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Advertisements
Solution
Let I = `int (3x - 2)/((x + 1)^2(x + 3))*dx`
Let `(3x - 2)/((x + 1)^2(x + 3)) = "A"/(x + 1) + "B"/(x + 1)^2 + "C"/(x + 3)`
∴ 3x – 2 = A(x + 1)(x + 3) + B(x + 3) + C(x + 1)2
Put x + 1 = 0, i.e. x = – 1, we get
– 3 – 2 = A(0)(2) + B(2) + C(0)
∴ – 5 = 2B
∴ B = `-(5)/(2)`
Put x + 3 = 0, i.e. x - – 3, we get
– 9 – 2 = A(– 2)(0) + B(0) + C(– 2)2
∴ – 11 = 4C
∴ C = `-(11)/(4)`
Put x = 0, we get
– 2 = A(1)(3) + B(3) + C(1)
∴ – 2 = 3A + 3B + C
∴ – 2 = `3"A" - (15)/(2) - (11)/(4)`
∴ 3A = `-2 + (15)/(2) + (11)/(4)`
= `(-8 + 30 + 11)/(4)`
∴ A = `(11)/(4)`
∴ `(3x - 2)/((x + 1)^2(x + 3)) = ((11/4))/(x + 1) + ((-5/4))/(x + 1)^2 + ((-11/4))/(x + 3)`
∴ I = `int [((11/4))/(x + 1) + (((-5)/2))/(x + 1)^2 + (((-11)/4))/(x + 3)]`
= `(11)/(4) int 1/(x + 1)*dx - (5)/(2) int (x + 1)^-2*dx - (11)/(4) int 1/(x + 3)*dx`
= `(11)/(4)log|x + 1| -(5)/(2)*(x + 1)^-1/(-1)*(1)/(1)- (11)/(4)log|x + 3| + c`
= `(11)/(4)log|(x + 1)/(x + 3)| + (5)/(2(x + 1)) + c`.
APPEARS IN
RELATED QUESTIONS
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 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:
`((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:
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int sqrt(4^x(4^x + 4)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int sec^3x "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int xcos^3x "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/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)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)
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 = ______.
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`
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
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
Which expression defines a rational function?
How are the constants \[\mathrm{A},\mathrm{B},\mathrm{C},\ldots\] determined in a partial-fraction decomposition?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
