Advertisements
Advertisements
Question
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Advertisements
Solution
Let I = `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`= int "e"^"x" [((2 + "x") - 1)/(2 + "x")^2]` dx
`= int "e"^"x" [1/(2 + "x") - 1/(2 + "x")^2]`dx
Let f(x) = `1/(2 + "x")`
∴ f '(x) = `(-1)/(2 +"x")^2`
∴ I = `int "e"^"x" ["f"("x") + "f" '("x")]` dx
`= "e"^"x" * "f"("x") + "c"`
`= "e"^"x" * 1/(2 + "x")` + c
∴ I = `"e"^"x"/(2 + "x")` + c
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`sqrt(sin 2x) cos 2x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
`int logx/(log ex)^2*dx` = ______.
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
