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`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
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 : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
`int "dx"/(("x" - 8)("x" + 7))`=
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sin(logx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 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` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "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 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^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 = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
When is a rational function called improper?
Integration by partial fractions is a method used to integrate which functions?
A proper rational function can be expressed as a sum of simpler rational functions called what?
Which partial form corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x-\mathrm{b})(x-\mathrm{c})}\]?
Which partial-fraction decomposition is used for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})^{2}}\]?
What should be checked before beginning partial-fraction decomposition?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
What must be included for a repeated linear factor?
