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
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
`intx^2 e^(x^3) dx` equals:
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int x^3 e^(x^2)`dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
`int 1/sqrt(2x^2 - 5) "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int e^(logcosx)dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate `int(1 + x + x^2/(2!))dx`.
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
