Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `e^(3x)/(e^(3x) + 1)`
Advertisements
Solution
Let I = `int e^(3x)/(e^(3x) + 1).dx`
Put e3x + 1 = t.
∴ 3e3x dx = dt
∴ e3x dx = `dt/(3)`
∴ I = `int (1)/t.dt/(3)`
= `(1)/(3) int (1)/t dt`
= `(1)/(3)log|t| + c`
= `(1)/(3)log|e^(3x) + 1| + c`.
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`1/(1 - tan x)`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Evaluate: `int 1/(sqrt("x") + "x")` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int (f^'(x))/(f(x))dx` = ______ + c.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate.
`int (5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
