Advertisements
Advertisements
Question
Evaluate: `int "e"^sqrt"x"` dx
Advertisements
Solution
Let I = `int "e"^sqrt"x"` dx
Put `sqrt"x"` = t
∴ x = t2
∴ dx = 2t dt
∴ I = `int "e"^"t" * "2t"`dt
`= 2 int "t" * "e"^"t" * "dt"`
`= 2 ["t" int "e"^"t" "dt" - int {"d"/"dx" ("t") int "e"^"t" * "dt"}"dt"]`
`= 2 ["t" * "e"^"t" - int 1 * "e"^"t" "dt"]`
`= 2("te"^"t" - "e"^"t")` + c
`= 2"e"^"t" ("t - 1")` + c
∴ I = `2"e"^sqrt"x" (sqrt"x" - 1)` + c
APPEARS IN
RELATED QUESTIONS
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Write a value of
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
`int sqrt(1 + "x"^2) "dx"` =
Evaluate `int (5"x" + 1)^(4/9)` dx
Evaluate: ∫ |x| dx if x < 0
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int sin^-1 x`dx = ?
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluate the following.
`int 1/(x^2+4x-5) dx`
`int x^3 e^(x^2) dx`
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
