Advertisements
Advertisements
Question
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Advertisements
Solution
Let I = `int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Let `(3"e")^"2t" + 5 = "A"(4"e"^"2t" - 5) + "B" "d"/"dt" (4"e"^"2t" - 5)`
`= 4 "Ae"^"2t" - 5"A" + "B"(8"e"^"2t")`
∴ `(3"e")^"2t" + 5 = (4"A" + 8"B") "e"^"2t" - 5"A"`
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"^"2t" - 5) + 7/8 (8"e"^"2t"))/(4"e"^"2t" - 5)` dt
`= - int "dt" + 7/8 int (8"e"^"2t")/(4"e"^"2t" - 5)` dt
∴ I = `- "t" + 7/8 log |4"e"^"2t" - 5|` + c .....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
`int "dx"/(9"x"^2 + 1)= ______. `
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int cos sqrtx` dx = _____________
`int sin^-1 x`dx = ?
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate `int1/(x(x - 1))dx`
