Advertisements
Advertisements
Question
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Advertisements
Solution
Let I = `int 1/(1 + "e"^"x")`dx
Dividing Nr. and Dr. by ex, we get
I = `int "e"^-"x"/("e"^-"x" + 1)` dx
Put `"e"^-"x" + 1` = t
∴ `- "e"^-"x" "dx" = "dt"`
∴ `"e"^-"x" "dx" = - "dt"`
∴ I = `int (- "dt")/"t" = - log |"t"| + "c"`
∴ I = - log `|"e"^-"x" + 1|` + c
APPEARS IN
RELATED QUESTIONS
Integrate the function in x sin x.
Integrate the function in x log x.
Integrate the function in (sin-1x)2.
Integrate the function in (x2 + 1) log x.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
`int ("x" + 1/"x")^3 "dx"` = ______
`int 1/(4x + 5x^(-11)) "d"x`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int ("d"x)/(x - x^2)` = ______
Evaluate `int 1/(x log x) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int1/(x+sqrt(x)) dx` = ______
`inte^(xloga).e^x dx` is ______
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
