Integrate the following functions w.r.t. x : ∫14-5cosx.dx - Mathematics and Statistics

Advertisements
Advertisements
Sum

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

Advertisements

Solution

Let I = `int (1)/(4 - 5cosx).dx`

Put `tan(x/2)` = t
∴ x = 2 tan–1 t

∴ dx = `(2dt)/(1 + t^2) and cosx = (1 - t^2)/(1 + t^2)`

∴ I = `int (1)/(4 - 5((1 - t^2)/(1 + t^2))).(2dt)/(1 + t^2)`

= `int (1 + t^2)/(4(1 + t^2) - 5(1 - t^2)).(2dt)/(1 + t^2)`

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

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

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

= `(2)/(9) xx (1)/(2 xx 1/3) log|(t - 1/3)/(t + 1/3)| + c`

= `(1)/(3) log |(3tan(x/2) - 1)/(3tan (x/2) + 1)| + c`.

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

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Exercise 3.2 (B) | Q 2.2 | Page 123

RELATED QUESTIONS

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`


Evaluate :`intxlogxdx`


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`


Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.


Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`


 
 

Evaluate :

`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`

 
 

Integrate the functions:

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


Integrate the functions:

`1/(x + x log x)`


Integrate the functions:

sin x ⋅ sin (cos x)


Integrate the functions:

`sqrt(ax + b)`


Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 


Integrate the functions:

`x^2/(2+ 3x^3)^3`


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

`1/(cos^2 x(1-tan x)^2`


Integrate the functions:

`sqrt(sin 2x) cos 2x`


Integrate the functions in `1/(1 - tan x)`


Integrate the functions in `((x+1)(x + logx)^2)/x`


Choose the correct answer int `(10x^9 + 10^x log_e 10)/(x^10 + 10^x)` dx equals

(A) 10x – x10 + C

(B) 10x + x10 + C

(C) (10x – x10)–1 + C

(D) log (10x + x10) + C


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`


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

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

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

Write a value of

\[\int\frac{1 + \cot x}{x + \log \sin x} \text{ dx }\]

Write a value of

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

Write a value of

\[\int e^x \left( \sin x + \cos x \right) \text{ dx}\]

 


Write a value of

\[\int \tan^6 x \sec^2 x \text{ dx }\] .

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\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]


Write a value of\[\int\frac{\cos x}{\sin x \log \sin x} dx\]

 


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\[\int e^{ax} \cos\ bx\ dx\].

 


\[\int\frac{\cos^5 x}{\sin x} \text{ dx }\]

\[\int x \sin^3 x\ dx\]

Integrate the following w.r.t. x : `3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`


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 sqrt(1 + sin 2x) dx`


Evaluate the following integrals:

`int x/(x + 2).dx`


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


Evaluate the following integrals:

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


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


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


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


Integrate the following functions w.r.t. x : sin4x.cos3x


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 : `(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:

`(5 - 3x)(2 - 3x)^(-1/2)`


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


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


Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`


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


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


Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`


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


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

cos8xcotx


Integrate the following functions w.r.t. x : tan5x


Integrate the following functions w.r.t. x : cos7x


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 (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)/sqrt(2x^2 - 5).dx`


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


Evaluate the following : `int  (1)/(x^2 + 8x + 12).dx`


Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`


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


Evaluate the following:

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


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


Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).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`


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


Evaluate the following integrals:

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


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


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


Evaluate the following : `int (logx)2.dx`


Choose the correct option from the given alternatives : 

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


Choose the correct options from the given alternatives :

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


Choose the correct options from the given alternatives :

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


Choose the correct options from the given alternatives : 

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


Choose the correct options from the given alternatives :

`int (cos2x - 1)/(cos2x + 1)*dx` =


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


Evaluate the following.

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


Evaluate the following.

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


Evaluate the following.

`int 1/(sqrt(3"x"^2 - 5))` dx


Evaluate:

`int (5x^2 - 6x + 3)/(2x − 3)` dx


Evaluate `int (5"x" + 1)^(4/9)` dx


Evaluate `int 1/((2"x" + 3))` dx


Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).


Evaluate: `int 1/(sqrt("x") + "x")` dx


Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx


`int sqrt(x^2 + 2x + 5)` dx = ______________


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


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


`int (sin4x)/(cos 2x) "d"x`


`int logx/x  "d"x`


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


`int sqrt(x)  sec(x)^(3/2) tan(x)^(3/2)"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)`


Evaluate `int(3x^2 - 5)^2  "d"x`


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


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


`int1/(4 + 3cos^2x)dx` = ______ 


`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.


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


`int ("d"x)/(x(x^4 + 1))` = ______.


`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.


`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.


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


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 `sqrt(2) int (sinx  dx)/(sin(x - π/4))` is ______.


Write `int cotx  dx`.


`int (logx)^2/x dx` = ______.


Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.


Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.


Evaluated the following

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


Evaluate the following.

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


if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`


Evaluate the following

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


Evaluate.

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


Evaluate the following.

`int 1/(x^2 + 4x - 5)  dx`


Evaluate `int1/(x(x - 1))dx`


Evaluate the following.

`int x sqrt(1 + x^2)  dx`


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


Evaluate the following.

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


`int "cosec"^4x  dx` = ______.


Evaluate.

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


Evaluate the following.

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


Evaluate:

`intsqrt(3 + 4x - 4x^2)  dx`


`int (cos4x)/(sin2x + cos2x)dx` = ______.


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


Evaluate `int1/(x(x-1))dx` 


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


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


Share
Notifications



      Forgot password?
Use app×