Advertisements
Advertisements
प्रश्न
`int 1/(sinx(3 + 2cosx)) "d"x`
Advertisements
उत्तर
Let I = `int 1/(sinx(3 + 2cosx)) "d"x`
= `int (sin x "d"x)/(sin^2x(3 + 2cosx))`
= `int (sin x "d"x)/((1 - cos^2x)(3 + 2cos x))`
= `int (sin x "d"x)/((1 + cos x)(1 - cos x)(3 + 2cos x))`
Put cos x = t
∴ − sin x d x = dt
∴ I = `int (-1)/((1 + "t")(1 - "t")(3 + 2"t")) "dt"`
Let `1/((1 + "t")(1 - "t")(3 + 2"t"))`
= `"A"/(1 + "t") + "B"/(1 - "t") + "C"/(3 + 2"t")`
∴ −1 = A(1 − t)(3 + 2t) + B(1 + t)(3 + 2t) + C(1 + t)(1 − t) .......(i)
Putting t = 1 in (i), we get
−1 = 10B
∴ B = `(-1)/10`
Putting t = −1 in (i), we get
−1 = 2A
∴ A = `(-1)/2`
Putting t = `-3/2` in (i), we get
−1 = `-5/4 "C"
∴ C = `4/5`
∴ `(-1)/((1 + "t")(1 - "t")(3 + 2"t")) = ((-1)/2)/(1 + "t") + ((-1)/10)/(1 - "t") + ((-4)/5)/(3 + 2"t")`
∴ I = `int[(-1)/(2(1 + "t")) + ((-1))/(10(1 - "t")) + 4/(5(3 + 2"t"))] "dt"`
= `-1/2 int 1/(1 + "t") "dt" - 1/10 int 1/(1 - "t") * "dt" + 4/5 int 1/(3 + 2"t") "dt"`
= `(-1)/2 log|1 + "t"| - 1/10 * (log|1 - "t"|)/(-1) + 4/5 * (log|3 + 2"t"|)/2 + "c"`
∴ I = `(-1)/2 log|1 + cos x| + 1/10 log|1 - cos x| + 2/5 log|3 + 2cos x| + "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:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`1/(x(x^4 - 1))`
`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 :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
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)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
`int "dx"/(("x" - 8)("x" + 7))`=
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int 1/(2 + cosx - sinx) "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 x/((x - 1)^2 (x + 2)) "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/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
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 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)
`int 1/(x^2 + 1)^2 dx` = ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + 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 (5x^2 - 6x + 3)/(2x - 3)dx`
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
Which partial form is appropriate for \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})^{2}(x-\mathrm{b})}\]?
Which decomposition corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\]?
What should be checked before beginning partial-fraction decomposition?
Which factorisation is correct for \[x^{2}-5x+6\]?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
What numerator should be used for each distinct linear factor in a partial-fraction decomposition?
