Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Solution
Let I = `int (4e^x - 25)/(2e^x - 5).dx`
Put,
Numerator = `"A (Denominator) + B"[d/dx ("Denominator")]`
∴ 4ex – 25 = `"A"(2e^x - 5) + "B"[d/dx(2e^x - 5)]`
= A(2ex – 5) + B(2ex – 0)
∴ 4ex – 25 = (2A + 2B)ex – 5A
Equating the coefficient of ex and constant on both sides, we get
2A + 2B = 4 ...(1)
and
5A = 25
∴ A = 5
∴ from (1),2(5) + 2B = 4
∴ 2B = – 6
∴ B = – 3
∴ 4ex – 25 = 5(2ex – 5) – 3 (2ex)
∴ I = `int[(5(2e^xx - 5) - 3(2e^x))/(2e^x - 5)].dx`
= `int[5 - (3(2e^x))/(2e^x - 5)].dx`
= `5 int 1dx - 3 int (2e^x)/(2e^x - 5].dx`
= 5x – 3 log|2ex – 5| + c ...`[∵ int (f'(x))/f(x)dx = log|f(x)| + c]`
APPEARS IN
RELATED QUESTIONS
Find : `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of
Write a value of
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate: `int "e"^sqrt"x"` dx
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int (log x)/(log ex)^2` dx = _________
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int x^x (1 + logx) "d"x`
`int cot^2x "d"x`
`int cos^7 x "d"x`
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int sec^6 x tan x "d"x` = ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int(1+ x + x^2/(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`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
`int x^3 e^(x^2) dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate:
`int sqrt((a - x)/x) dx`
`int 1/(sin^2x cos^2x)dx` = ______.
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
