Advertisements
Advertisements
प्रश्न
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
Advertisements
उत्तर
Let I = `int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
= `int (sin 2x)/(3(sin^2x)^2 - 4sin^2x + 1) "d"x`
Put sin2x = t
∴ 2 sin x cos x dx = dt
∴ sin 2x dx = dt
∴ I = `int "dt"/(3"t"^2 - 4"t" + 1)`
= `int "dt"/(3("t"^2 - 4/3"t" + 1/3)`
`(1/2 "coefficient of " "t")^2 = [1/2 xx ((-4)/3)]^2 = 4/9`
∴ I = `1/3 int 1/("t"^2 - 4/3"t" + 4/9 - 4/9 + 1/3) "dt"`
= `1/3 int 1/(("t"^2 - 4/3"t" + 4/9) - 1/9) "dt"`
= `1/3 int 1/(("t" - 2/3)^2 - (1/3)^2) "dt"`
= `1/3*1/(2 xx 1/3) log|(("t" - 2/3) - 1/3)/(("t" - 2/3) + 1/3)| + "c"`
= `1/2 log|(3"t" - 3)/(3"t" - 1)| + "c"`
∴ I = `1/2 log|(3sin^2x - 3)/(3sin^2x - 1)| + "c"`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`1/(x^4 - 1)`
`int (dx)/(x(x^2 + 1))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
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)/(sin2x + cosx)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
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 with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
`int 1/(x(x^3 - 1)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int sqrt((9 + x)/(9 - x)) "d"x`
`int sec^3x "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`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
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 "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 = ______.
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 (x + 7)/(x^2 + 4x + 7)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
When is a rational function \[\frac{P(x)}{Q(x)}\] called proper?
Integration by partial fractions is a method used to integrate which functions?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
What is done after long division, before writing the appropriate partial-fraction decomposition?
Which factorisation is correct for \[x^{2}-5x+6\]?
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
