Advertisements
Advertisements
Question
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
Advertisements
Solution
Let I = `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`= int (5"x"^2 + 20"x" + 6)/("x"("x"^2 + 2"x" + 1))` dx
`= int (5"x"^2 + 20"x" + 6)/("x"("x + 1")^2)` dx
Let `(5"x"^2 + 20"x" + 6)/("x"("x + 1")^2) = "A"/"x" + "B"/"x + 1" + "C"/("x + 1")^2`
∴ 5x2 + 20x + 6 = A(x + 1)2 + B(x + 1)x + Cx ...(i)
Putting x = 0 in (i), we get
5(0) + 20(0) + 6 = A(1)2 + B(1)(0) + C(0)
∴ A = 6
Putting x = - 1 in (i), we get
5 (1) + 20(- 1) + 6 = A (0)+ B (0) (- 1) + C (-1)
∴ - 9 = - C
∴ C = 9
Putting x = 1 in (i), we get
5 (1) + 20 (1) + 6 = A (2)2 + B (2) (1) + C (1)
∴ 31 = 4A + 2B + C
∴ 31 = 4(6) + 2B + 9
∴ B = - 1
∴ `(5"x"^2 + 20"x" + 6)/("x"("x + 1")^2) = 6/"x" + (-1)/"x + 1" + 9/("x + 1")^2`
∴ I = `int [6/"x" + (- 1)/"x + 1" + 9/("x + 1")^2]` dx
`= 6 int 1/"x" "dx" - int 1/"x + 1" "dx" + 9 int ("x + 1")^-2` dx
`= 6 log |"x"| - log |"x + 1"| + 9("x + 1")^-1/(-1)` + c
∴ I = `6 log |"x"| - log |"x + 1"| - 9/("x + 1")` + c
APPEARS IN
RELATED QUESTIONS
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
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 : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
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 : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int x^7/(1 + x^4)^2 "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`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`
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"`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
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)
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)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
Integration by partial fractions is a method used to integrate which functions?
What numerator is used for an irreducible quadratic factor?
