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
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int cos^7 x "d"x`
If f'(x) = `x + 1/x`, then f(x) is ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
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 the following.
`int 1/(x^2+4x-5) dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate `int1/(x(x-1))dx`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
