Advertisements
Advertisements
प्रश्न
Evaluate: `int "e"^sqrt"x"` dx
Advertisements
उत्तर
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
संबंधित प्रश्न
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`(1+ log x)^2/x`
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
`int logx/(log ex)^2*dx` = ______.
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
`int secx/(secx - tanx)dx` equals ______.
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
What is integration by substitution?
Which expression results after simplifying \[\int\frac{\sin(t-a)}{\sin t}\,dt\]?
What is \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
