Evaluate: ∫(tanx+cotx)dx - Mathematics and Statistics

Advertisements
Advertisements
Sum

Evaluate:

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

Advertisements

Solution

I = `int(sqrt(tanx) + 1/sqrt(tanx))dx`

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

Put `sqrt(tanx)` = t

∴ tan x = t2

∴ x = tan–1t2

∴ 1 dx = `1/(1 + (t^2)^2) 2t  dt`

∵ sec2x dx = 2t dt

∴ dx = `(2t)/(sec^2x)dx`

= `(2t)/(1 + tan^2x)dx`

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

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

Put `t - 1/t` = u

∵ `[d/dt(t - 1/t) = 1 + 1/t^2]`

∴ `(t - (-1/t^2))dt` = 1 du

∴ `(1 + 1/t^2)dt` = 1 du

I = `2int1/(u^2 + 2)du`

= `2int1/(u^2 + (sqrt(2))^2)du`

= `2 1/sqrt(2) tan^-1(u/sqrt(2)) + c`

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

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

= `sqrt(2) tan^-1 ((tanx - 1)/(sqrt(2)*sqrt(tanx))) + c`

  Is there an error in this question or solution?

RELATED QUESTIONS

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 :   `∫1/(cos^4x+sin^4x)dx`


Integrate the functions:

`sqrt(ax + b)`


Integrate the functions:

`1/(x-sqrtx)`


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

`cos x /(sqrt(1+sinx))`


Integrate the functions:

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


`(10x^9 + 10^x log_e 10)/(x^10 + 10^x)  dx` equals:


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`


\[\int\sqrt{3 + 2x - x^2} \text{ dx}\]

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

\[\int\cos x \sqrt{4 - \sin^2 x}\text{ dx}\]

Write a value of

\[\int \tan^3 x \sec^2 x \text{ dx }\].

 


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

 


Write a value of\[\int \cos^4 x \text{ sin x dx }\]


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


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


Write a value of

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

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


Write a value of

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

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\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].


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


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

 

 


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


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


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

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

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


 Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`


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


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


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


Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`


Evaluate the following integrals:

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


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


Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`


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


Evaluate the following integrals:

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


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 : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`


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 : `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 : sin4x.cos3x


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


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


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

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


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 : `(x^n - 1)/sqrt(1 + 4x^n)`


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


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 : `sin(x - a)/cos(x  + b)`


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


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


Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`


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


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


Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`


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


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


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


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` =


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


Choose the correct options from the given alternatives :

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


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


Evaluate the following.

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


Evaluate the following.

∫ (x + 1)(x + 2)7 (x + 3)dx


Evaluate the following.

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


Fill in the Blank.

`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______


Evaluate: ∫ |x| dx if x < 0


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


Evaluate: `int log ("x"^2 + "x")` dx


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


`int ("e"^(3x))/("e"^(3x) + 1)  "d"x`


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


`int 1/(xsin^2(logx))  "d"x`


`int (cos2x)/(sin^2x)  "d"x`


`int cot^2x  "d"x`


`int x/(x + 2)  "d"x`


`int(log(logx))/x  "d"x`


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


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:

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


State whether the following statement is True or False:

`int"e"^(4x - 7)  "d"x = ("e"^(4x - 7))/(-7) + "c"`


Evaluate `int(3x^2 - 5)^2  "d"x`


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


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


`int sec^6 x tan x   "d"x` = ______.


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


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


`int(7x - 2)^2dx = (7x -2)^3/21 + c`


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/sqrt(1 - 2x^4) dx` = ______.

(where c is a constant of integration)


`int dx/(2 + cos x)` = ______.

(where C is a constant of integration)


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


Evaluate the following.

`int 1/(x^2+4x-5)  dx`


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


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


Solve the following Evaluate.

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


Evaluate the following

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


Evaluate:

`int sqrt((a - x)/x) dx`


If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)


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


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


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


Evaluate the following.

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


Evaluate:

`intsqrt(3 + 4x - 4x^2)  dx`


Evaluate the following.

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


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


Evaluate the following.

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


Share
Notifications



      Forgot password?
Use app×