Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
उत्तर
Let I = `int (3e^(2x) + 5)/(4e^(2x) - 5).dx`
Put,
Numerator = `"A (Denominator) + B"[d/dx("Denominator")]`
∴ 3e2x + 5 = `"A"(4e^(2x) - 5) + "B"[d/dx(4e^(2x) - 5)]`
= A(4e2x – 5) + B(4.e2x x 2 – 0)
∴ 3e2x + 5 = (4A + 8B)e2x – 5A
Equating the coeffiecient of e2x and constant on both sides, we get
4A + 8B = 3 ...(1)
and
– 5A = 5
∴ A = – 1
∴ from (1), 4(– 1) + 8B = 3
∴ 8B = 7
∴ B = `(7)/(8)`
∴ 3e2x + 5 = `- (4e^(2x) - 5) + 7/8(8e^(2x))`
∴ I = `int[(-(4e^(2x) - 5) +7/8(8e^(2x)))/(4e^(2x) - 5)].dx`
= `int[-1 +(7/8(8e^(2x)))/(4e^(2x) - 5)].dx`
= `int 1 dx + (7)/(8) int (8e^(2x))/(4e^(2x) - 5).dx`
= `- x + (7)/(8)log|4e^(2x) - 5| + c ...[∵ int (f'(x))/f(x)dx = log|f(x)| + c]`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`1/(1 + cot x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Write a value of
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals : `int(4x + 3)/(2x + 1).dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`
Integrate the following functions w.r.t.x:
cos8xcotx
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
`int sqrt(1 + "x"^2) "dx"` =
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Evaluate: `int 1/(sqrt("x") + "x")` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int cos^7 x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
Evaluate `int(3x^2 - 5)^2 "d"x`
`int sin^-1 x`dx = ?
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int sec^6 x tan x "d"x` = ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
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 ______.
Write `int cotx dx`.
Evaluated 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"`
if `f(x) = 4x^3 - 3x^2 + 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)
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
`int "cosec"^4x dx` = ______.
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
