Integrate the following functions w.r.t. x : e2x+1e2x-1 - Mathematics and Statistics

Advertisements
Advertisements
Sum

Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`

Advertisements

Solution

Let I = `int (e^(2x) + 1)/(e^(2x) - 1).dx`

= `int (((e^(2x) + 1)/(e^x)))/(((e^(2x) - 1)/(e^x))).dx`

= `int((e^x + e^(-x))/(e^x - e^-x)).dx`

= `int (d/dx(e^x - e^-x))/(e^x - e^-x).dx`

= log|ex – e–x| + c.    ...`[∵ int (f'(x))/f(x).dx= log|f(x)| + c]`

  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.2 (A) [Page 110]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Exercise 3.2 (A) | Q 1.08 | Page 110

RELATED QUESTIONS

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`


Show that:  `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`


Evaluate : `int(x-3)sqrt(x^2+3x-18)  dx`


Evaluate :

`int(sqrt(cotx)+sqrt(tanx))dx`


Find : `int((2x-5)e^(2x))/(2x-3)^3dx`


Find : `int(x+3)sqrt(3-4x-x^2dx)`


Find `intsqrtx/sqrt(a^3-x^3)dx`


Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`


Evaluate: `int sqrt(tanx)/(sinxcosx) dx`


Integrate the functions:

`1/(x + x log x)`


Integrate the functions:

sin x ⋅ sin (cos x)


Integrate the functions:

(4x + 2) `sqrt(x^2 + x +1)`


Integrate the functions:

`(x^3 - 1)^(1/3) x^5`


Integrate the functions:

`x/(e^(x^2))`


Integrate the functions:

`(e^(2x) - 1)/(e^(2x) + 1)`


Integrate the functions:

tan2(2x – 3)


Integrate the functions:

`(sin^(-1) x)/(sqrt(1-x^2))`


Integrate the functions:

`cos sqrt(x)/sqrtx`


Evaluate : `∫1/(3+2sinx+cosx)dx`


Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`


Evaluate: `int (2y^2)/(y^2 + 4)dx`


Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`


Evaluate: `int (sec x)/(1 + cosec x) dx`


\[\int\sqrt{x - x^2} dx\]

\[\int\sqrt{1 + x - 2 x^2} \text{ dx }\]

\[\int\sqrt{9 - x^2}\text{ dx}\]

Write a value of

\[\int x^2 \sin x^3 \text{ dx }\]

Write a value of

\[\int e^x \sec x \left( 1 + \tan x \right) \text{ dx }\]

 Write a valoue of \[\int \sin^3 x \cos x\ dx\]

 


Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]


Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].


Write a value of\[\int \log_e x\ dx\].

 


Write a value of\[\int a^x e^x \text{ dx }\]


Write a value of\[\int\left( e^{x \log_e \text{  a}} + e^{a \log_e x} \right) dx\] .


Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]


Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]


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\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .


Evaluate:  \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]


\[\text{ If } \int\left( \frac{x - 1}{x^2} \right) e^x dx = f\left( x \right) e^x + C, \text{ then  write  the value of  f}\left( x \right) .\]

\[If \int e^x \left( \tan x + 1 \right)\text{ sec  x  dx } = e^x f\left( x \right) + C, \text{ then  write  the value  of  f}\left( x \right) .\]

 

 


\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]


 Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log  |"x" +sqrt("x"^2 +"a"^2) | + "c"`


Integrate the following w.r.t. x : `2x^3 - 5x + 3/x + 4/x^5`


Evaluate the following integrals : tan2x dx


Evaluate the following integrals : `int sin x/cos^2x dx`


Evaluate the following integrals:

`int (cos2x)/sin^2x dx` 


Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`


Evaluate the following integrals : `int tanx/(sec x + tan x)dx`


Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`


Evaluate the following integrals : `int sin 4x cos 3x dx`


Evaluate the following integrals : `int(4x + 3)/(2x + 1).dx`


Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`


Evaluate the following integrals:

`int (sin4x)/(cos2x).dx`


Evaluate the following integrals:

`int(2)/(sqrt(x) - sqrt(x + 3)).dx`


Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`


If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)


Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`


Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`


Integrate the following functions w.r.t. x : `(1)/(4x + 5x^-11)`


Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1 


Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`


Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`


Integrate the following functions w.r.t. x : `(10x^9  10^x.log10)/(10^x + x^10)`


Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`


Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`


Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`


Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`


Integrate the following functions w.r.t. x : `sin(x - a)/cos(x  + b)`


Integrate the following functions w.r.t.x:

cos8xcotx


Integrate the following functions w.r.t. x : sin5x.cos8x


Evaluate the following : `int (1)/(25 - 9x^2).dx`


Evaluate the following : `int (1)/(7 + 2x^2).dx`


Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`


Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`


Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`


Evaluate the following : `int sinx/(sin 3x).dx`


Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`


Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`


Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`


Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`


Evaluate the following integrals:

`int (7x + 3)/sqrt(3 + 2x - x^2).dx`


Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`


Evaluate the following integrals : `int (3cosx)/(4sin^2x + 4sinx - 1).dx`


Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`


Choose the correct options from the given alternatives :

`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =


`int logx/(log ex)^2*dx` = ______.


Choose the correct options from the given alternatives :

`int (e^(2x) + e^-2x)/e^x*dx` =


Evaluate `int (3"x"^2 - 5)^2` dx


If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).


If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).


Evaluate the following.

`int 1/("x"^2 + 4"x" - 5)` dx


Evaluate the following.

`int 1/(4"x"^2 - 20"x" + 17)` dx


Evaluate the following.

`int 1/(7 + 6"x" - "x"^2)` dx


Evaluate the following.

`int 1/(sqrt(3"x"^2 + 8))` dx


Choose the correct alternative from the following.

The value of `int "dx"/sqrt"1 - x"` is


`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?


Choose the correct alternative from the following.

`int "dx"/(("x" - "x"^2))`= 


Fill in the Blank.

`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c


If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______


Evaluate: `int "x" * "e"^"2x"` dx


`int 1/sqrt((x - 3)(x - 2))` dx = ________________


`int x^2/sqrt(1 - x^6)` dx = ________________


`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________


`int 2/(sqrtx - sqrt(x + 3))` dx = ________________


`int cos sqrtx` dx = _____________


`int (log x)/(log ex)^2` dx = _________


`int  ("e"^x(x - 1))/(x^2)  "d"x` = ______ 


`int logx/x  "d"x`


`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`


`int ("e"^(2x) + "e"^(-2x))/("e"^x)  "d"x`


`int (cos2x)/(sin^2x)  "d"x`


`int cos^7 x  "d"x`


State whether the following statement is True or False:

If `int x  "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`


State whether the following statement is True or False:

`int sqrt(1 + x^2) *x  "d"x = 1/3(1 + x^2)^(3/2) + "c"`


`int sin^-1 x`dx = ?


If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.


`int dx/(1 + e^-x)` = ______


`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`


`int (cos x)/(1 - sin x) "dx" =` ______.


If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.


`int sec^6 x tan x   "d"x` = ______.


`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.


If f'(x) = `x + 1/x`, then f(x) is ______.


If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.


`int (sin  (5x)/2)/(sin  x/2)dx` is equal to ______. (where C is a constant of integration).


`int(log(logx) + 1/(logx)^2)dx` = ______.


`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.


The value of `intsinx/(sinx - cosx)dx` equals ______.


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 ______.


The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.


The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.


If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.


Evaluate `int(1 + x + x^2/(2!) )dx`


Evaluate `int(1 + x + x^2/(2!))dx`


Evaluate the following.

`int x sqrt(1 + x^2)  dx`


Prove that:

`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.


`int x^3 e^(x^2) dx`


Evaluate:

`int 1/(1 + cosα . cosx)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`


`int "cosec"^4x  dx` = ______.


Evaluate:

`int sin^2(x/2)dx`


`int 1/(sin^2x cos^2x)dx` = ______.


Evaluate:

`int(cos 2x)/sinx dx`


Evaluate the following.

`intxsqrt(1+x^2)dx`


Evaluate the following.

`int (x^3)/(sqrt(1 + x^4)) dx`


Evaluate the following:

`int x^3/(sqrt(1+x^4))dx`


Evaluate the following

`int x^3 e^(x^2) ` dx


Evaluate `int (1 + "x" + "x"^2/(2!))`dx


Evaluate.

`int (5x^2 -6x + 3)/(2x -3)dx`


Share
Notifications



      Forgot password?
Use app×