Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
उत्तर
Let I = `int (x^2 + x - 1)/(x^2 + x - 6).dx`
= `int ((x^2 + x - 6) + 5)/(x^2 + x - 6).dx`
= `int [1 + (5)/(x^2 + x - 6)].dx`
= `int 1 dx + 5 int (1)/(x^2 + x - 6).dx`
Let `(1)/(x^2 + x - 6)`
= `(1)/((x + 3)(x - 2)`
= `"A"/(x + 3) + "B"/(x- 2)`
∴ 1 = A(x – 2) + B(x + 3)
Put x 3 = 0, i.e. x = –3, we get
1 = A(– 5) + B(0)
∴ A = `(-1)/(5)`
Put x – 2 = 0, i.e. x = 2, we get
1 = A(0) + B(5)
∴ B = `(1)/(5)`
∴ `(1)/(x^2 + x - 6) = ((-1/5))/(x + 3) + ((1/5))/(x - 2)`
∴ I = `int 1 dx + 5 int [((-1/5))/(x + 3) + ((1/5))/(x - 2)].dx`
= `int 1 dx - int (1)/(x + 3).dx + int (1)/(x - 2).dx`
= x – log|x + 3| + log|x – 2| + c
= `x + log|(x - 2)/(x + 3)| + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`1/(x^4 - 1)`
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`
Find :
`∫ sin(x-a)/sin(x+a)dx`
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(x^5 + 1)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
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)/(sinx*(3 + 2cosx)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
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)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int (2"x" + 1)/("x"("x - 1")("x - 4"))` dx
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
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 = ______
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`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"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 ______.
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.
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)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 `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
