D∫dxsinxcosx+2cos2x = ______. - Mathematics

Advertisements
Advertisements
MCQ
Fill in the Blanks

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

Options

  • `log |cos x + 2| + "c"`

  • `log |sin x + 2| + "c"`

  • `log |tan x + 2| + "c"`

  • `log |tanx - 2| + "c"`

Advertisements

Solution

`int ("d"x)/(sinx cosx + 2cos^2x)` = `log |tan x + 2| + "c"`.

Explanation:

Let I = `int ("d"x)/(sinx cosx + 2cos^2x)`

= `int (sec^2x)/(tan x + 2) "d"x`

Put tan x + 2 = t

⇒ sec2x dx = dt

∴ I = `int "dt"/"t" = log|"t"| + "c"`

= `log|tanx + 2| + "c"`

  Is there an error in this question or solution?

RELATED QUESTIONS

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`


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


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`


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


Evaluate :   `∫1/(cos^4x+sin^4x)dx`


 
 

Evaluate :

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

 
 

Integrate the functions:

(4x + 2) `sqrt(x^2 + x +1)`


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

`(sin^(-1) x)/(sqrt(1-x^2))`


Integrate the functions:

`sqrt(sin 2x) cos 2x`


Integrate the functions:

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


Evaluate : `∫1/(3+2sinx+cosx)dx`


Evaluate `int 1/(3+ 2 sinx + 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`


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

\[\int\sqrt{16 x^2 + 25} \text{ dx}\]

\[\int\sqrt{2 x^2 + 3x + 4} \text{ dx}\]

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

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

 


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 value of

\[\int e^{\text{ log  sin x  }}\text{ cos x}. \text{ dx}\]

Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].


Write a value of

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

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


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


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


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


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


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

`int "dx"/(9"x"^2 + 1)= ______. `


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


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


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


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


Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`


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


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


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


Evaluate the following integrals:

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


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


Evaluate the following integrals:

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


Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`


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


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 : e3logx(x4 + 1)–1 


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:

`(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 : `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)/(sinx.cosx + 2cos^2x)`


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


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


Integrate the following functions w.r.t. x : sin5x.cos8x


Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`


Evaluate the following : `int sqrt((9 + x)/(9 - 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)/sqrt(x^2 + 8x - 20).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)/(3 + 2sin x - cosx)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`


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


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


Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).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 (cos2x - 1)/(cos2x + 1)*dx` =


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


If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).


Evaluate the following.

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


Evaluate the following.

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


Choose the correct alternative from the following.

`int "x"^2 (3)^("x"^3) "dx"` =


Evaluate:

`int (5x^2 - 6x + 3)/(2x − 3)` dx


Evaluate `int 1/((2"x" + 3))` dx


Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx


Evaluate: `int sqrt(x^2 - 8x + 7)` dx


`int 1/sqrt((x - 3)(x - 2))` dx = ________________


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


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


`int 2/(sqrtx - sqrt(x + 3))` 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 x/(x + 2)  "d"x`


`int cos^7 x  "d"x`


`int(log(logx))/x  "d"x`


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


Choose the correct alternative:

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


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)`


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


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


`int (cos x)/(1 - sin x) "dx" =` ______.


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


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


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


`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"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 ______.


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


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


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


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


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


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


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


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


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


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


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


Evaluate the following

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


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


Evaluate the following.

`intx sqrt(1 +x^2)  dx`


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


Evaluate:

`int sin^3x cos^3x  dx`


The value of `int dx/(sqrt(1 - x))` is ______.


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


Share
Notifications



      Forgot password?
Use app×