Advertisements
Advertisements
प्रश्न
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
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
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
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:
`(x^3 + x + 1)/(x^2 -1)`
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))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
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 : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x : `(x + 5)/(x^3 + 3x^2 - x - 3)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int (2"x" + 1)/("x"("x - 1")("x - 4"))` dx
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int x^7/(1 + x^4)^2 "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int 1/(4x^2 - 20x + 17) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sin(logx) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int ("d"x)/(2 + 3tanx)`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
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`
`int x/((x - 1)^2 (x + 2)) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
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"`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin 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_-2^1 sqrt(5 - 4x - x^2)dx`
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)
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 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
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`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
