Advertisements
Advertisements
Question
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
Solution
Let I = `int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
= `int sqrt(("e"^(2x)("e"^x - 1))/("e"^x + 1)) "d"x`
= `int"e"^x sqrt(("e"^x - 1)/("e"^x + 1)) "d"x`
Put ex = t
∴ ex dx = dt
∴ I = `int sqrt(("t" - 1)/("t" + 1)) "dt"`
= `int sqrt(("t" - 1)/("t" + 1) xx ("t" - 1)/("t" - 1)) "dt"`
= `int ("t" - 1)/sqrt("t"^2 - 1) "dt"`
= `int ("t"/sqrt("t"^2 - 1) - 1/sqrt("t"^2 - 1)) "dt"`
= `int "t"/sqrt("t"^2 - 1) "dt" - int 1/sqrt("t"^2 - 1) "dt"`
= I1 − I2 .......(i)
I1 = `int "t"/sqrt("t"^2 - 1) "dt"`
Put t2 − 1 = a
∴ 2t dt = da
∴ I1 = `1/2 int "da"/sqrt("a")`
= `1/2 int "a"^(1/2) "da"`
= `1/2("a"^(1/2)/(1/2)) + "c"_1`
= `sqrt("a") + "c"_1`
= `sqrt("t"^2 - 1) + "c"_1`
∴ I1 = `sqrt("e"^(2x) - 1) + "c"_1` ......(ii)
I2 = `int 1/sqrt("t"^2 - 1^2) "dt"`
= `log|"t" + sqrt("t"^2 - 1^2)| + "c"_2`
∴ I2 = `log|"e"^x + sqrt("e"^(2x) - 1)| + "c"_2` .......(iiii)
From (i), (ii) and (iii), we get
I = `sqrt("e"^(2x) - 1) - log|"e"^x + sqrt("e"^(2x) - 1)| +"c"`,
where c = c1 − c2
APPEARS IN
RELATED QUESTIONS
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`1/(1 + cot x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Solve: dy/dx = cos(x + y)
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
`int "dx"/(9"x"^2 + 1)= ______. `
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x : x3 + x2 – x + 1
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : sin5x.cos8x
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
`int logx/(log ex)^2*dx` = ______.
Integrate the following w.r.t.x : `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
`int 1/(cos x - sin x)` dx = _______________
`int sqrt(1 + sin2x) "d"x`
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
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).
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate:
`int sin^2(x/2)dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate:
`int(cos 2x)/sinx dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
