Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Advertisements
Solution
Let I = `int (20 + 12e^x)/(3e^x + 4).dx`
∴ 20 + 12ex = `"A"(3e^x + 4) + "B"d/dx(3e^x + 4)`
= 3Aex + 4A + 3Bex
∴ 20 + 12ex = 4A + (3A + 3B) ex
By Equating the coefficient of on both sides, we get
4A = 20 and 3A + 3B = 12
Solving these equations, we get
A = 5 and B = - 1
∴ 20 + 12ex = 5(3ex + 4) – 3ex
∴ I = `int(5(3e^x + 4) - 3e^x)/(3e^x + 4) dx`
= `5 intdx - int (3e^x)/(3e^x + 4] dx`
∴ `"I" = 5x - log|3e^x + 4| + c` ... [∵ `int(f'(x))/f(x)dx = log |f (x)| + c`]
APPEARS IN
RELATED QUESTIONS
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`sin x/(1+ cos x)`
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
`int sqrt(1 + sin2x) dx`
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int x/(x + 2) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int1/(4 + 3cos^2x)dx` = ______
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
`int cos^3x dx` = ______.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
In \[\int\sin^3x\cos^2x\,dx\], which rewriting prepares the integrand for the substitution \[t=\cos x\]?
