Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Advertisements
उत्तर
Let I = `int log (1 + x)^((1 + x)).dx`
= `int (1 + x)log(1 + x).dx`
= `int [log(1 + x)] (1 + x).dx`
= `[log(1 + x) int (1 + x).dx - int[d/dt {log(1 + x)} int (1 + x).dx].dx`
= `[log (1 + x)] [(1 + x)^2/2] - int 1/(x + 1).(x + 1)^2/(2).dx`
= `(x + 1)^2/(2).log(1 + x) - (1)/(2) int (x + 1).dx`
= `(x + 1)^2/(2).log (1 + x) - (1)/(2).(x + 1)^2/(2) + c`
= `(x + 1)^2/(2)[log (1 + x) - 1/2] + c`.
APPEARS IN
संबंधित प्रश्न
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate : sec3 x w. r. t. x.
Integrate the function in x sin x.
Integrate the function in x sin 3x.
Integrate the function in (x2 + 1) log x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following: `int x.sin^-1 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 : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (cos2x)/(sin^2x cos^2x) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
Evaluate `int 1/(x(x - 1)) "d"x`
∫ log x · (log x + 2) dx = ?
`int log x * [log ("e"x)]^-2` dx = ?
Find `int_0^1 x(tan^-1x) "d"x`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate the following.
`int x^3 e^(x^2) dx`
`int1/(x+sqrt(x)) dx` = ______
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`int logx dx = x(1+logx)+c`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate:
`int x^2 cos x dx`
The value of `inta^x.e^x dx` equals
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
