Advertisements
Advertisements
Question
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Advertisements
Solution
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
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
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(x^5 + 1)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int "dx"/(("x" - 8)("x" + 7))`=
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sin(logx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x sin2x cos5x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "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 = ______
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^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate`int(5x^2-6x+3)/(2x-3)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
