English

Evaluate the following integrals : ∫1+sin2xdx - Mathematics and Statistics

Advertisements
Advertisements

Question

Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`

Sum

Solution

`int sqrt(1 + sin 2x) dx`

= `intsqrt(cos^2x + sin^2x + 2sin x cos x) dx`

= `intsqrt((cos x + sin x)^2)dx`

= `int(cos x + sinx)dx`

= `int cos x  dx + int sin x  dx`

= sin x – cos x + c.

shaalaa.com
  Is there an error in this question or solution?
Chapter 3: Indefinite Integration - Exercise 3.1 [Page 102]

APPEARS IN

RELATED QUESTIONS

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


Find : `int((2x-5)e^(2x))/(2x-3)^3dx`


 
 

Evaluate :

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

 
 

Evaluate: `int sqrt(tanx)/(sinxcosx) dx`


Integrate the functions:

`x/(sqrt(x+ 4))`, x > 0 


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

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


Integrate the functions:

`1/(1 + cot x)`


Evaluate `int 1/(3+ 2 sinx + cosx) dx`


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

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

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


Write a value of

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

Write a value of

\[\int\frac{1 + \log x}{3 + x \log x} \text{ dx }\] .

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

 


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


The value of \[\int\frac{1}{x + x \log x} dx\] is


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


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


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


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


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


Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`


Integrate the following function 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:

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


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


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


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


Evaluate the following : `int sqrt((10 + x)/(10 - x)).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)/(2sin 2x - 3)dx`


Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).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 dx/(cosxsqrt(sin^2x - cos^2x))*dx` =


Integrate the following with respect to the respective variable:

`x^7/(x + 1)`


Evaluate `int (3"x"^2 - 5)^2` dx


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


Evaluate the following.

`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx


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


Evaluate the following.

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


Evaluate: ∫ |x| dx if x < 0


Evaluate: `int 1/(2"x" + 3"x" log"x")` dx


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


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


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


`int cos^7 x  "d"x`


State whether the following statement is True or False:

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


`int1/(4 + 3cos^2x)dx` = ______ 


`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "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 ("d"x)/(x(x^4 + 1))` = ______.


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


The value of `intsinx/(sinx - cosx)dx` equals ______.


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


Evaluate the following.

`int 1/(x^2 + 4x - 5)  dx`


Evaluate the following.

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


Evaluate the following

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


`int "cosec"^4x  dx` = ______.


Evaluate:

`int sin^2(x/2)dx`


`int x^2/sqrt(1 - x^6)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 `int1/(x(x - 1))dx`


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


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


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×