Advertisements
Advertisements
Question
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Advertisements
Solution
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
RELATED QUESTIONS
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in tan-1 x.
Integrate the function in `e^x (1/x - 1/x^2)`.
Evaluate the following : `int x^3.tan^-1x.dx`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
`int (sinx)/(1 + sin x) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
Evaluate `int 1/(x(x - 1)) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int 1/sqrt(x^2 - 9) dx` = ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`intsqrt(1+x) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3e^(x^2) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int x^2 cos x dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
