Advertisements
Advertisements
प्रश्न
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Advertisements
उत्तर
Let I = `int "e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`= int e^x/sqrt((e^x)^2 + 4e^x + 13)` dx
Put ex = t
∴ ex dx = dt
∴ I = `(dt)/(sqrt(t^2 + 4t + 13))`
`= int 1/sqrt(t^2 + 4t + 4 - 4 + 13)` dt
`= int 1/(sqrt((t + 2)^2 + 9))` dt
`= int 1/(sqrt((t + 2)^2 + (3)^2))` dt
`= log |t + 2 + sqrt((t + 2)^2 + (3)^2)|` + c
`= log |(t + 2) + sqrt(t^2 + 4t + 13)| + c`
∴ I = `log |(e^x + 2) + sqrt(e^(2x) + 4e^x + 13)| + c`
APPEARS IN
संबंधित प्रश्न
Integrate the function in x sin 3x.
Integrate the function in x log x.
Integrate the function in x sin−1 x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int ("d"x)/(x - x^2)` = ______
`int(x + 1/x)^3 dx` = ______.
`int"e"^(4x - 3) "d"x` = ______ + c
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
`inte^(xloga).e^x dx` is ______
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`inte^x sinx dx`
Evaluate:
`int (logx)^2 dx`
Evaluate the following:
`intx^3e^(x^2)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate:
`int x^2 cos x dx`
Which function is an example under priority \(I\) in the LIATE rule?
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
