English

Evaluate the following. ∫14x2-1 dx - Mathematics and Statistics

Advertisements
Advertisements

Question

Evaluate the following.

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

Sum
Advertisements

Solution

Let I = `int "dx"/(4"x"^2 - 1)`

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

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

`= 1/4 xx 1/(2 xx 1/2) log |("x" - 1/2)/("x" + 1/2)|` + c

∴ I = `1/4` log `|("2x" - 1)/("2x" + 1)|` + c

Alternate Method:

Let I = `int "dx"/(4"x"^2 - 1) = int "dx"/((2"x"^2) - (1)^2)`

`= 1/(2 xx 1) xx 1/2 log |("2x" - 1)/("2x" + 1)|` + c

∴ I = `1/4` log `|("2x" - 1)/("2x" + 1)|` + c

shaalaa.com
  Is there an error in this question or solution?
Chapter 5: Integration - EXERCISE 5.4 [Page 129]

RELATED QUESTIONS

Integrate the functions:

`1/(x + x log x)`


Integrate the functions:

`x/(9 - 4x^2)`


Integrate the functions:

tan2(2x – 3)


Integrate the functions:

sec2(7 – 4x)


Integrate the functions:

`1/(1 - tan x)`


Evaluate `int 1/(3+ 2 sinx + cosx) dx`


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


Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]


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

 


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


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


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


Integrate the following functions w.r.t. x :  tan 3x tan 2x tan x


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


Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`


Choose the correct options from the given alternatives :

`int (e^x(x - 1))/x^2*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`.


`int (cos2x)/(sin^2x)  "d"x`


State whether the following statement is True or False:

`int3^(2x + 3)  "d"x = (3^(2x + 3))/2 + "c"`


`int x^3"e"^(x^2) "d"x`


General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)


`int(sin2x)/(5sin^2x+3cos^2x)  dx=` ______.


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 ______.


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


Find `int (x + 2)/sqrt(x^2 - 4x - 5) 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`


Evaluate the following.

`int1/(x^2 + 4x-5)dx`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×