Advertisements
Advertisements
प्रश्न
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
उत्तर
Let I = `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`
= `int_(logsqrt(2))^(logsqrt(3)) 1/(((e^(2x) + 1))/e^x xx ((e^(2x) - 1))/e^x) dx`
= `int_(logsqrt(2))^(logsqrt(3)) e^(2x)/((e^(4x) - 1))dx`
Let e2x = t
Then, 2e2x dx = dt
= `int_2^3 dt/(2(t^2 - 1))`
= `1/2 int_2^3 dt/(t^2 - 1^2)`
= `[1/2 xx 1/(2 xx 1) log|(t - 1)/(t + 1)|]_2^3`
= `1/4 [log ((3 - 1)/(3 + 1)) - log ((2 - 1)/(2 + 1))]`
= `1/4 [log 2/4 - log 1/3]`
= `1/4 [log 1/2 + log 3]`
= `1/4 [log 1/2 xx 3]`
= `1/4 log 3/2`.
APPEARS IN
संबंधित प्रश्न
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
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 sqrt("x"^2 + 2"x" + 5)` dx
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int cos sqrtx` dx = _____________
Evaluate `int(3x^2 - 5)^2 "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
