Advertisements
Advertisements
Question
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Advertisements
Solution
Let I = `int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Let 20 - 12ex = A(3ex - 4) + B `"d"/"dx"`(3ex - 4)
= 3 Aex - 4A + 3Bex
∴ 20 - 12ex = (3A + 3B)ex - 4A
Comparing the coefficients of ex and constant term on both sides, we get
- 4A = 20 and 3A + 3B = - 12
Solving these equations, we get
A = -5 and B = 1
∴ I = `int (-5(3"e"^"x" - 4) + 3"e"^"x")/(3"e"^"x" - 4)`dx
`= - 5 int "dx" + int (3"e"^"x")/(3"e"^"x" - 4)` dx
∴ I = - 5x + log `|(3"e"^"x" - 4)|` + c ....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
Notes
The answer in the textbook is incorrect.
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of\[\int \log_e x\ dx\].
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Integrate the following functions w.r.t. x : tan5x
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int(1 - x)^(-2) dx` = ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
If f'(x) = 4x3- 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int1/(x(x - 1))dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
