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cos4x/(sqrtsin4x+ one)
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Similar expressions
cos4x/(sqrtsin4x-1)

Integral of cos4x/(sqrtsin4x+1) dx

The solution

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  1                    
  /                    
 |                     
 |      cos(4*x)       
 |  ---------------- dx
 |    __________       
 |  \/ sin(4*x)  + 1   
 |                     
/                      
0

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

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

Detail solution

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

$- \frac{\log{\left(\sqrt{\sin{\left(4 x \right)}} + 1 \right)}}{2} + \frac{\sqrt{\sin{\left(4 x \right)}}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(\sqrt{\sin{\left(4 x \right)}} + 1 \right)}}{2} + \frac{\sqrt{\sin{\left(4 x \right)}}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                              
 |                             __________      /      __________\
 |     cos(4*x)              \/ sin(4*x)    log\1 + \/ sin(4*x) /
 | ---------------- dx = C + ------------ - ---------------------
 |   __________                   2                   2          
 | \/ sin(4*x)  + 1                                              
 |                                                               
/

$$\int \frac{\cos{\left(4 x \right)}}{\sqrt{\sin{\left(4 x \right)}} + 1}\, dx = C - \frac{\log{\left(\sqrt{\sin{\left(4 x \right)}} + 1 \right)}}{2} + \frac{\sqrt{\sin{\left(4 x \right)}}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ________      /      ________\
\/ sin(4)    log\1 + \/ sin(4) /
---------- - -------------------
    2                 2

$$- \frac{\log{\left(1 + \sqrt{\sin{\left(4 \right)}} \right)}}{2} + \frac{\sqrt{\sin{\left(4 \right)}}}{2}$$

  ________      /      ________\
\/ sin(4)    log\1 + \/ sin(4) /
---------- - -------------------
    2                 2

$$- \frac{\log{\left(1 + \sqrt{\sin{\left(4 \right)}} \right)}}{2} + \frac{\sqrt{\sin{\left(4 \right)}}}{2}$$

sqrt(sin(4))/2 - log(1 + sqrt(sin(4)))/2

Numerical answer [src]

(-0.140934640209814 + 0.0771497634534624j)

(-0.140934640209814 + 0.0771497634534624j)

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

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Identical expressions

Similar expressions

Integral of cos4x/(sqrtsin4x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2