Advertisements
Advertisements
Question
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
Advertisements
Solution
Let I = `int "e"^"x"/(4"e"^"2x" -1)` dx
`"I" = int "e"^"x"/(4("e"^"x")^2 - 1)` dx
Put ex = t
∴ ex dx = dt
∴ I = `int "dt"/(4"t"^2 - 1)`
`∴ "I" = 1/4 int 1/("t"^2 - 1/4)` dt
`∴ "I" = 1/4 int 1/("t"^2 - (1/2)^2)` dt
`∴ "I" = 1/4 . 1/(2 (1/2)) log |("t" - 1/2)/("t" + 1/2)|` + c
`∴ "I" = 1/4 log |("2t" - 1)/("2t" + 1)|` + c
Resubstitute t = ex
`∴ "I" = 1/4 log |(2"e"^"x" - 1)/(2"e"^"x" + 1)|` + c
Notes
Answer in the textbook is incorrect.
APPEARS IN
RELATED QUESTIONS
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate the function in (sin-1x)2.
Evaluate the following: `int logx/x.dx`
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
Evaluate `int 1/(x log x) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate `int tan^-1x dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) 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)
Evaluate the following.
`intx^3 e^(x^2)dx`
Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?
Under the LIATE rule, which type of function has priority \(L\)?
