Advertisements
Advertisements
प्रश्न
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Advertisements
उत्तर
Let `sin x = t => cos x dx = dt`
`int (2dt)/((1-t)(1+t^2))`
Using partial fraction
`2/((1-t)(1+t^2)) = A/((1-t)) + (Bt + C)/((1+t^2))`
On solving A = 1, B =1, C = 1
`int (2dt)/((1-t)(1+t^2)) = int (dt)/((1-t)) + int ((1+t))/((1+t^2)) dt`
`= int (dt)/((1-t)) + int (dt)/(1+t^2) + int (tdt)/((1+t^2))`
`= -In (1-t) + tan^(-1) t + 1/2 In (1+t^2)`
`= In sqrt(1+t^2)/(1-t) + tan^(-1) t + C`
Replacing the value of t
`= In sqrt(1+sin^2x)/(1-sinx) + tan^(-1)(sin x) + C`
APPEARS IN
संबंधित प्रश्न
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
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 : `(1)/(x(1 + 4x^3 + 3x^6)`
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 : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
`int "dx"/(("x" - 8)("x" + 7))`=
`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 sin(logx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int 1/(sinx(3 + 2cosx)) "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
Evaluate `int x log x "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)
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
A proper rational function can be expressed as a sum of simpler rational functions called what?
What is done after long division, before writing the appropriate partial-fraction decomposition?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
