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 x.
Integrate the function in x log 2x.
Integrate the function in (x2 + 1) log x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
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 the following.
`int x^3 e^(x^2)`dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/(5 - 16"x"^2)`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Evaluate:
`int (logx)^2 dx`
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Evaluate `int tan^-1x dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
`∫ sin^(−1)` xdx is equal to ______.
