Advertisements
Advertisements
प्रश्न
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Advertisements
उत्तर
Let `I = int (2x)/((x^2 + 1)(x^2 + 3))` dx
Putting x2 = t, 2x dx = dt
`therefore I = int dt/((t + 1)(t + 3))`
Now, `1/((t + 1)(t + 3)) = A/(t + 1) = B/(t + 3)`
1 = A(t + 3) + B(t + 1)
Put t = -1
1 = A(-1 + 3)
⇒ 1 = 2A
∴ A `= 1/2`
Put t = -3
1 = B (-3 + 1)
⇒ 1 = -2B
∴ B `= -1/2`
`therefore 1/((t + 1)(t + 3)) = 1/(2(t + 1)) - 1/(2(t + 3))`
`therefore I = int 1/((t + 1)(t + 3)) dt = 1/2 int 1/(t + 1) dt - 1/2 int 1/(t + 3) dt`
`= 1/2 log (t + 1) - 1/2 log (t + 3) + C`
`= 1/2 log abs ((t + 1)/(t + 3)) + C`
`= 1/2 log abs ((x^2 + 1)/(x^2 + 3)) + C`
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 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))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
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 with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`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 ("d"x)/(2 + 3tanx)`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int xcos^3x "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
Which pair of equations is obtained by equating coefficients in \[5x-5=\mathrm{A}(x-3)+\mathrm{B}(x-2)\]?
What are the values of \[\mathrm{A}\] and \[\mathrm{B}\] in \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\]?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
