Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
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)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Evaluate `int (5"x" + 1)^(4/9)` dx
Evaluate: `int 1/(sqrt("x") + "x")` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int (cos2x)/(sin^2x) "d"x`
`int(1 - x)^(-2) dx` = ______.
`int (7x + 9)^13 "d"x` ______ + c
`int(5x + 2)/(3x - 4) dx` = ______
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int1/(x(x - 1))dx`
