Evaluate : ∫(√cotx+√tanx)dx - Mathematics

Advertisements
Advertisements

Evaluate :

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

Advertisements

Solution

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

`=int(sqrt(tanx)(1+cotx))dx`

`Let tanx=t^2`

Differentiating both sides w.r.t. x, we get

`sec^2 x dx=2t dt`

`=> dx=(2tdt)/(1+t^4)`

`therefore I=intt(1+1/t^2)xx(2t)/(1+t^4)dt`

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

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

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

`Let (t−1)/t=y`

`=>(1+1/t^2)dt=dy`

`therefore I=2int 1/(y^2+(sqrt2)^2) dy`

`=2xx1/sqrt2 tan^-1(y/sqrt2)+C`

`=sqrt2 tan^-1 (t-1/t)/sqrt2+C`

`=sqrt2 tan^-1 ((t^2-1)/(sqrt2t))+C`

`=sqrt2 tan^-1((tanx-1)/sqrt(2tanx))+C`

 

  Is there an error in this question or solution?
2014-2015 (March) Patna Set 2

RELATED QUESTIONS

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


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


Evaluate :`intxlogxdx`


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


Find : `int((2x-5)e^(2x))/(2x-3)^3dx`


Find : `int(x+3)sqrt(3-4x-x^2dx)`


Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`


Find `intsqrtx/sqrt(a^3-x^3)dx`


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


Integrate the functions:

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


Integrate the functions:

tan2(2x – 3)


Integrate the functions:

`(2cosx - 3sinx)/(6cos x + 4 sin x)`


Integrate the functions:

`sin x/(1+ cos x)`


Integrate the functions:

`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`


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`


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

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

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

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

Write a value of

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

Write a value of

\[\int\frac{\cos x}{3 + 2 \sin x}\text{  dx}\]

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


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


Write a value of

\[\int\frac{a^x}{3 + a^x} \text{ dx}\]

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


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


Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].


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


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

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


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


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


Find : ` int  (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`


Integrate the following w.r.t. x : x3 + x2 – x + 1


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


Evaluate the following integrals:

`int (cos2x)/sin^2x dx` 


Evaluate the following integrals : `int sinx/(1 + sinx)dx`


Evaluate the following integrals : `int tanx/(sec x + tan x)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 : `intsqrt(1 + sin 5x).dx`


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


Evaluate the following integrals:

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


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


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


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


Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`


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


Integrate the following functions w.r.t. x : x9.sec2(x10)


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


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


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


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


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


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


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


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


Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`


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 : `3^(cos^2x) sin 2x`


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


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 sqrt((10 + x)/(10 - x)).dx`


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


Evaluate the following : `(1)/(4x^2 - 20x + 17)`


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


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


Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`


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


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


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


Choose the correct options from the given alternatives :

`int sqrt(cotx)/(sinx*cosx)*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 :

`int f x^x (1 + log x)*dx`


Choose the correct options from the given alternatives :

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


integrate the following with respect to the respective variable : `x^2/(x + 1)`


Integrate the following w.r.t.x : `(3x + 1)/sqrt(-2x^2 + x + 3)`


Evaluate the following.

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


Evaluate the following.

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


Evaluate the following.

`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx


Evaluate the following.

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


`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


`int 1/(cos x - sin x)` dx = _______________


If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______


`int sqrt(1 + sin2x)  "d"x`


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


`int x/(x + 2)  "d"x`


`int(log(logx))/x  "d"x`


`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1))  "d"x`


`int (7x + 9)^13  "d"x` ______ + c


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"`


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


`int sin^-1 x`dx = ?


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


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


`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?


`int1/(4 + 3cos^2x)dx` = ______ 


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


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


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 ______.


`int (x + sinx)/(1 + cosx)dx` is equal to ______.


`int sqrt(x^2 - a^2)/x dx` = ______.


`int cos^3x  dx` = ______.


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


`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`


Find `int dx/sqrt(sin^3x cos(x - α))`.


Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.


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


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


Evaluated 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.

`int 1/(x^2+4x-5)  dx`


Evaluate the following.

`int 1/(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.

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


Prove that:

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


Evaluate.

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


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


Evaluate the following.

`intx sqrt(1 +x^2)  dx`


Evaluate:

`int sin^3x cos^3x  dx`


Evaluate the following.

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


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


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


Evaluate the following.

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


Share
Notifications



      Forgot password?
Use app×