Advertisements
Advertisements
Question
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Advertisements
Solution
Let I = `int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Let 3e2t + 5 = `"A"(4"e"^(2"t") - 5) + "B" "d"/"dt"(4"e"^(2"t") - 5)`
= 4Ae2t – 5A + B(8e2t)
∴ 3e2t + 5 = (4A + 8B) e2t – 5A
Comparing the coefficients of e2t and constant term on both sides,
we get 4A + 8B = 3 and – 5A = 5
Solving these equations,
we get A = – 1 and B = `7/8`
∴ I = `int (-1(4"e"^(2"t") - 5) + 7/8(8"e"^(2"t")))/(4"e"^(2"t") - 5) "dt"`
= `/int "dt" + 7/8 int (8"e"^(2"t"))/(4"e"^(2"t") - 5) "dt"`
∴ I = `-"t" + 7/8 log|4"e"^(2"t") - 5| + "c"` ......`[because int ("f'"(x))/("f"(x)) "d"x = log|"f"(x)| + "c"]`
APPEARS IN
RELATED QUESTIONS
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) 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)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
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.
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int xcos^3x "d"x`
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"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 ______.
Evaluate: `int (dx)/(2 + cos x - sin x)`
`int 1/(x^2 + 1)^2 dx` = ______.
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 = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
