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
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Integrate the functions:
sec2(7 – 4x)
Write a value of
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 sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Evaluate: `int "e"^sqrt"x"` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int(1 - x)^(-2) dx` = ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
