Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Advertisements
उत्तर
Let I = `int ("e"^"x" + "e"^(- "x"))^2 * ("e"^"x" - "e"^(-"x"))`dx
Put `"e"^"x" + "e"^(- "x")` = t
∴ `"e"^"x" - "e"^(- "x")`dx = dt
∴ I = `int "t"^2 . "dt"`
∴ I = `(t^(2 + 1))/(2 + 1) + c`
∴ I = `(t^3)/(3) + c`
∴ I = `[("e"^"x" + "e"^(- "x"))^3]/(3) + c`
APPEARS IN
संबंधित प्रश्न
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Evaluate the following integrals:
tan2x dx
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate: `int "e"^sqrt"x"` dx
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int (log x)/(log ex)^2` dx = _________
`int logx/x "d"x`
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
If f'(x) = `x + 1/x`, then f(x) is ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
`int secx/(secx - tanx)dx` equals ______.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
