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Identical expressions
cos(4x)/ two ^(three *sin(4x))
co sinus of e of (4x) divide by 2 to the power of (3 multiply by sinus of (4x))
co sinus of e of (4x) divide by two to the power of (three multiply by sinus of (4x))
cos(4x)/2(3*sin(4x))
cos4x/23*sin4x
cos(4x)/2^(3sin(4x))
cos(4x)/2(3sin(4x))
cos4x/23sin4x
cos4x/2^3sin4x
cos(4x) divide by 2^(3*sin(4x))
cos(4x)/2^(3*sin(4x))dx

Solving integrals/ cos(4x)/2^(3*sin(4x))

Integral of cos(4x)/2^(3*sin(4x)) dx

The solution

You have entered [src]

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

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

Integral(cos(4*x)/(2^(3*sin(4*x))), (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 $\frac{du}{4}$ :
  
  $\int \frac{2^{- 3 \sin{\left(u \right)}} \cos{\left(u \right)}}{16}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2^{- 3 \sin{\left(u \right)}} \cos{\left(u \right)}}{4}\, du = \frac{\int 2^{- 3 \sin{\left(u \right)}} \cos{\left(u \right)}\, du}{4}$
    1. Let $u = 2^{- 3 \sin{\left(u \right)}}$ .
      
      Then let $du = - 3 \cdot 2^{- 3 \sin{\left(u \right)}} \log{\left(2 \right)} \cos{\left(u \right)} du$ and substitute $- \frac{du}{3 \log{\left(2 \right)}}$ :
      
      $\int \frac{1}{9 \log{\left(2 \right)}^{2}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{3 \log{\left(2 \right)}}\right)\, du = - \frac{\int 1\, du}{3 \log{\left(2 \right)}}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $- \frac{u}{3 \log{\left(2 \right)}}$
      
      Now substitute $u$ back in:
      
      $- \frac{2^{- 3 \sin{\left(u \right)}}}{3 \log{\left(2 \right)}}$
    So, the result is: $- \frac{2^{- 3 \sin{\left(u \right)}}}{12 \log{\left(2 \right)}}$
  Now substitute $u$ back in:
  
  $- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}$
Method #2
1. Let $u = \frac{1}{2^{3 \sin{\left(4 x \right)}}}$ .
  
  Then let $du = - 12 \cdot 2^{- 3 \sin{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(4 x \right)} dx$ and substitute $- \frac{du}{12 \log{\left(2 \right)}}$ :
  
  $\int \frac{1}{144 \log{\left(2 \right)}^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{12 \log{\left(2 \right)}}\right)\, du = - \frac{\int 1\, du}{12 \log{\left(2 \right)}}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    So, the result is: $- \frac{u}{12 \log{\left(2 \right)}}$
  Now substitute $u$ back in:
  
  $- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}$
Method #3
1. Let $u = 2^{3 \sin{\left(4 x \right)}}$ .
  
  Then let $du = 12 \cdot 2^{3 \sin{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(4 x \right)} dx$ and substitute $\frac{du}{12 \log{\left(2 \right)}}$ :
  
  $\int \frac{1}{144 u^{2} \log{\left(2 \right)}^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{12 u^{2} \log{\left(2 \right)}}\, du = \frac{\int \frac{1}{u^{2}}\, du}{12 \log{\left(2 \right)}}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
    So, the result is: $- \frac{1}{12 u \log{\left(2 \right)}}$
  Now substitute $u$ back in:
  
  $- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}$
Add the constant of integration:

$- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                       -3*sin(4*x)
 |   cos(4*x)           2           
 | ----------- dx = C - ------------
 |  3*sin(4*x)           12*log(2)  
 | 2                                
 |                                  
/

$$\int \frac{\cos{\left(4 x \right)}}{2^{3 \sin{\left(4 x \right)}}}\, dx = C - \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

             -3*sin(4)
    1       2         
--------- - ----------
12*log(2)   12*log(2)

$$- \frac{2^{- 3 \sin{\left(4 \right)}}}{12 \log{\left(2 \right)}} + \frac{1}{12 \log{\left(2 \right)}}$$

             -3*sin(4)
    1       2         
--------- - ----------
12*log(2)   12*log(2)

$$- \frac{2^{- 3 \sin{\left(4 \right)}}}{12 \log{\left(2 \right)}} + \frac{1}{12 \log{\left(2 \right)}}$$

Упростить

Numerical answer [src]

-0.459810210721908

-0.459810210721908

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of cos(4x)/2^(3*sin(4x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3