Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Advertisements
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 : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int \log_e x\ dx\].
Write a value of
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
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 - 1)^2)/(x^2 + 1)^2`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following : `int sinx/(sin 3x).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate `int 1/("x" ("x" - 1))` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int sqrt(1 + sin2x) dx`
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate `int1/(x(x - 1))dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int1/(x(x-1))dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
