English

Integrate the following functions w.r.t. x : ∫12sin2x-3dx

Advertisements
Advertisements

Question

Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`

Sum
Advertisements

Solution

Let I = `int (1)/(2sin 2x - 3)dx`

Put tan x = t
∴ x = tan–1 t

∴ dx = `dt/(1 + t^2) and sin 2x = (2t)/(1 + t^2)`

∴ I = `int(1)/(2((2t)/(1 + t^2)) - 3).dt/(1 + t^2)`

= `int (1 + t^2)/(4t - 3 - 3t^2).dt/(1 + t^2)`

= `int (1)/(-3t^2 + 4t - 3)dt`

= `(1)/(3) int (1)/(t^2 - 4/3t + 1)dt`

= `-(1)/(3)  int (1)/((t^2 - 4/3t + 4/9) - (4)/(9) + 1)dt`

= `-(1)/(3) int (1)/((t - 2/3)^2 + (sqrt(5)/3)^2)dt`

= `-(1)/(3) xx (1)/((sqrt(5)/3))tan^-1 ((t - 2/3)/(sqrt(5)/3)) + c`

= `-(1)/sqrt(5)tan^-1 ((3t - 2)/sqrt(5)) + c`

= `-(1)/sqrt(5)tan^-1((3tan x - 2)/(sqrt(5))) + c`.

shaalaa.com
  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.2 (B) [Page 123]

APPEARS IN

RELATED QUESTIONS

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`


Find : `int((2x-5)e^(2x))/(2x-3)^3dx`


Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`


Solve:

dy/dx = cos(x + y)


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


Write a value of\[\int e^{ax} \sin\ bx\ dx\]


The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is


Integrate the following w.r.t. x:

`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`


Evaluate the following integrals : `int (sin2x)/(cosx)dx`


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


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


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

`x^5sqrt(a^2 + x^2)`


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


Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`


Evaluate the following:

`int (1)/sqrt((x - 3)(x + 2)).dx`


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


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


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


Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx


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


Evaluate `int 1/(x (x - 1))` dx


Evaluate the following.

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


Evaluate the following.

`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt


Evaluate the following.

`int "x"^3/(16"x"^8 - 25)` dx


Choose the correct alternative from the following.

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


`int sqrt(1 + "x"^2) "dx"` =


State whether the following statement is True or False.

The proper substitution for `int x(x^x)^x (2log x + 1)  "d"x` is `(x^x)^x` = t


Evaluate `int (5"x" + 1)^(4/9)` dx


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


Evaluate: `int log ("x"^2 + "x")` dx


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


`int cos sqrtx` dx = _____________


`int sqrt(1 + sin2x)  dx`


To find the value of `int ((1 + logx))/x` dx the proper substitution is ______


`int (1 + x)/(x + "e"^(-x))  "d"x`


`int[ tan (log x) + sec^2 (log x)] dx= ` ______


If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.


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


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 ______.


`int cos^3x  dx` = ______.


`int secx/(secx - tanx)dx` equals ______.


Evaluate the following.

`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`


Evaluate the following.

`int x sqrt(1 + x^2)  dx`


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


Evaluate the following.

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


Evaluate:

`int(cos 2x)/sinx dx`


Evaluate:

`int sin^3x cos^3x  dx`


Evaluate the following.

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


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×