Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Advertisements
उत्तर
Let I = `int (1 + "x")/("x" + "e"^"-x")` dx
`= int (1 + "x")/("x" + 1/"e"^"x")` dx
`= int (1 + "x")/(("x" * "e"^"x" + 1)/"e"^"x")`dx
`= int ("e"^"x"(1 + "x"))/("x" * "e"^"x" + 1)` dx
Put `"x" * "e"^"x" + 1 = "t"`
∴ `["x" * ("e"^"x") + "e"^"x" (1) + 0]`dx = dt
∴ `"e"^"x" ("x" + 1)`dx = dt
∴ I = `int "dt"/"t"`
= log |t| + c
∴ I = log `|"x" * "e"^"x" + 1|` + c
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`sin x/(1+ cos x)`
Write a value of
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate: `int 1/(sqrt("x") + "x")` dx
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
`int x^3 e^(x^2) dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int sin^2(x/2)dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
