Advertisements
Advertisements
Question
`int "e"^x x/(x + 1)^2 "d"x`
Advertisements
Solution
Let I = `int "e"^x(x/((x + 1)^2))"d"x`
= `int"e"^x (((x + 1) - 1)/(x + 1)^2)"d"x`
= `int"e"^x ((x + 1)/(x + 1)^2 - 1/(x + 1)^2)"d"x`
= `int"e"^x (1/(x + 1) - 1/(x + 1)^2)"d"x`
Put f(x) = `1/(x + 1)`
∴ f'(x) = `(-1)/(x + 1)^2`
∴ I = `int"e"^x["f"(x) + "f'"(x)]"d"x`
= `"e"^x*"f"(x) + "c"`
∴ I = `"e"^x (1/(x + 1)) + "c"`
APPEARS IN
RELATED QUESTIONS
Integrate the function in x log 2x.
Integrate the function in `e^x (1/x - 1/x^2)`.
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int log(logx)/x.dx`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : e2x sin x cos x
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate:
∫ (log x)2 dx
`int (sinx)/(1 + sin x) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int(1-x)^-2 dx` = ______
`int logx dx = x(1+logx)+c`
`int(xe^x)/((1+x)^2) dx` = ______
Solve the following
`int_0^1 e^(x^2) x^3 dx`
The value of `inta^x.e^x dx` equals
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Under the LIATE rule, which type of function has priority \(L\)?
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
