Mister Exam

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Similar expressions
cos2x/sqrt(3-4sin2x)

Integral of cos2x/sqrt(3+4sin2x) dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |       cos(2*x)        
 |  ------------------ dx
 |    ________________   
 |  \/ 3 + 4*sin(2*x)    
 |                       
/                        
0

$$\int\limits_{0}^{1} \frac{\cos{\left(2 x \right)}}{\sqrt{4 \sin{\left(2 x \right)} + 3}}\, dx$$

Integral(cos(2*x)/sqrt(3 + 4*sin(2*x)), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\cos{\left(u \right)}}{2 \sqrt{4 \sin{\left(u \right)} + 3}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(u \right)}}{\sqrt{4 \sin{\left(u \right)} + 3}}\, du = \frac{\int \frac{\cos{\left(u \right)}}{\sqrt{4 \sin{\left(u \right)} + 3}}\, du}{2}$
    1. Let $u = 4 \sin{\left(u \right)} + 3$ .
      
      Then let $du = 4 \cos{\left(u \right)} du$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{1}{4 \sqrt{u}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{\sqrt{u}}\, du = \frac{\int \frac{1}{\sqrt{u}}\, du}{4}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$
      
      So, the result is: $\frac{\sqrt{u}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sqrt{4 \sin{\left(u \right)} + 3}}{2}$
    So, the result is: $\frac{\sqrt{4 \sin{\left(u \right)} + 3}}{4}$
  Now substitute $u$ back in:
  
  $\frac{\sqrt{4 \sin{\left(2 x \right)} + 3}}{4}$
Method #2
1. Let $u = \sqrt{4 \sin{\left(2 x \right)} + 3}$ .
  
  Then let $du = \frac{4 \cos{\left(2 x \right)} dx}{\sqrt{4 \sin{\left(2 x \right)} + 3}}$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{1}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    So, the result is: $\frac{u}{4}$
  Now substitute $u$ back in:
  
  $\frac{\sqrt{4 \sin{\left(2 x \right)} + 3}}{4}$
Add the constant of integration:

$\frac{\sqrt{4 \sin{\left(2 x \right)} + 3}}{4}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{4 \sin{\left(2 x \right)} + 3}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                               ________________
 |      cos(2*x)               \/ 3 + 4*sin(2*x) 
 | ------------------ dx = C + ------------------
 |   ________________                  4         
 | \/ 3 + 4*sin(2*x)                             
 |                                               
/

$$\int \frac{\cos{\left(2 x \right)}}{\sqrt{4 \sin{\left(2 x \right)} + 3}}\, dx = C + \frac{\sqrt{4 \sin{\left(2 x \right)} + 3}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___     ______________
  \/ 3    \/ 3 + 4*sin(2) 
- ----- + ----------------
    4            4

$$- \frac{\sqrt{3}}{4} + \frac{\sqrt{3 + 4 \sin{\left(2 \right)}}}{4}$$

    ___     ______________
  \/ 3    \/ 3 + 4*sin(2) 
- ----- + ----------------
    4            4

$$- \frac{\sqrt{3}}{4} + \frac{\sqrt{3 + 4 \sin{\left(2 \right)}}}{4}$$

-sqrt(3)/4 + sqrt(3 + 4*sin(2))/4

Numerical answer [src]

0.211055894381827

0.211055894381827

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of cos2x/sqrt(3+4sin2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2