Advertisements
Advertisements
Question
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
Solution
Let I = `int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
Put ex = t
⇒ exdx = dt
∴ I = `int 5/((t + 1)(t^2 + 9))dt`
Let `5/((t + 1)(t^2 + 9))`
= `A/(t + 1) + (Bt + C)/(t^2 + 9)`
∴ 5 = A(t2 + 9) + (Bt + C)(t + 1) ...(i)
Putting t = –1 in (i), we get
5 = A[(–1)2 + 9]
∴ 5 = 10A
∴ A = `1/2`
Putting t = 0 in (i), we get
5 = A(0 + 9) + (0 + C) (0 + 1)
∴ 5 = 9A + C
∴ 5 = `9(1/2) + "C"`
∴ C = `1/2`
Putting t = 1 in (i), we get
5 = A(12 + 9) + (B + C)(1 + 1)
∴ 5 = 10A + 2B + 2C
∴ 5 = `10(1/2) + 2B + 2(1/2)`
∴ – 1 = 2B
∴ B = `-1/2`
∴ `5/((t + 1)(t^2 + 9)) = (1/2)/(t + 1) + (1/2t + 1/2)/(t^2 + 9)`
∴ I = `int((1/2)/(t+ 1) + ((-1)/2t + 1/2)/(t^2 + 9)) dt`
= `1/2 [int 1/(t + 1) dt - int t/(t^2 + 9) dt + int 1/(t^2 + 9) dt]`
= `1/2[int 1/(t + 1) dt - 1/2 int (2t)/(t^2 + 9) dt + int 1/(t^2 + 3^2) dt]`
= `1/2 [log (t + 1) - 1/2 log (t^2 + 9) + 1/3tan^-1(t/3)] + c`
∴ I = `1/2 log (e^x + 1) - 1/4 log (e^(2x) + 9) + 1/6 tan^-1 ((e^x)/3) + c`
RELATED QUESTIONS
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`1/(x^4 - 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:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (dx)/(x(x^2 + 1))` equals:
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
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 : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
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 : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int (2"x" + 1)/("x"("x - 1")("x - 4"))` dx
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
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
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`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`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x sin2x cos5x "d"x`
`int 1/(sinx(3 + 2cosx)) "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 = ?
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x log x "d"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/(1 - x^4) "d"x` put x2 = t
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 ______.
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
Evaluate: `int (dx)/(2 + cos x - sin x)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
