Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
उत्तर
Let I = `int (5x^2 + 20x + 6)/(x^3 + 2x^2 + x)*dx`
= `int (5x^2 + 20x + 6)/(x(x^2 + 2x + 1))*dx`
= `int (5x^2 + 20x + 6)/(x(x + 1)^2)*dx`
Let `(5x^2 + 20x + 6)/(x(x + 1)^2) = "A"/x + "B"/(x + 1) + "C"/(x + 1)^2`
∴ 5x2 + 20x + 6 = A(x + 1)2 + Bx(x + 1) + Cx
Put x = 0, we get
0 + 0 + 6 = A(1) + B(0)(1) + C(0)
∴ A = 6
Put x + 1 = 0, i x = – 1, we get
5(1) + 20(– 1) + 6 = A(0) + B(– 1)(0) + C(– 1)
∴ – 9 = – C
∴ C = 9
Put x = 1, we get
5(1) + 20(1) + 6 = A(4) + B(1)(2) + C(1)
But A = 6 and C = 9
∴ 31 = 24 + 2B + 9
∴ B = – 1
∴ `(5x^2 + 20x + 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`
= `6log|x| - log|x + 1| + 9*((x + 1)^-1)/(-1) + c`
= `log|x^6| - log|x + 1| - (9)/((x + 1)) + c`
= `log|x^6/(x + 1)| - (9)/((x + 1)) + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`1/(x^2 - 9)`
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:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
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))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
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 : `(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 w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int (2"x" + 1)/("x"("x - 1")("x - 4"))` dx
Evaluate: `int "x"/(("x - 1")^2("x + 2"))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sec^3x "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int ("d"x)/(x^3 - 1)`
`int 1/(sinx(3 + 2cosx)) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
`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
Evaluate `int x^2"e"^(4x) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
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.
Find: `int x^2/((x^2 + 1)(3x^2 + 4))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 ______.
`int 1/(x^2 + 1)^2 dx` = ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
