Advertisements
Advertisements
Question
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Advertisements
Solution
`int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Putting ex = t and exdx = dt, we get
`int(e^x dx)/((e^x - 1)^2 (e^x + 2)) = int (dt)/((t-1)^2(t+2))`
Using partial fraction, we have
`1/((t-1)^2 (t + 1)) = A/(t-1)^2 + B/(t -1) + C/(t +2)`
⇒ 1 = A(t+2) + B(t−1)(t+2) + C(t−1)2 .....(1)
Putting t = 1 in (1), we get
`A = 1/3`
Putting t = −2 in (1), we get
C = `1/9`
Comparing the coefficients of t2 on both sides of (1), we get
B + C = 0
APPEARS IN
RELATED QUESTIONS
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
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)/(sin2x + cosx)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
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
`int x^2sqrt("a"^2 - x^6) "d"x`
`int 1/(2 + cosx - sinx) "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^3tan^(-1)x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
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)`
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 (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
When is a rational function called improper?
Integration by partial fractions is a method used to integrate which functions?
Which decomposition corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\]?
