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

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  • Integral of d{x}:
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  • Integral of x^2*e^(4*x) Integral of x^2*e^(4*x)
  • Integral of cos^4x Integral of cos^4x
  • Integral of sin(e^x) Integral of sin(e^x)

  • 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

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The solution

You have entered [src] 
  1               
  /               
 |                
 |    cos(4*x)    
 |  ----------- dx
 |   3*sin(4*x)   
 |  2             
 |                
/                 
0
∫01cos⁡(4x)23sin⁡(4x) dx\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

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫2−3sin⁡(u)cos⁡(u)16 du\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:

        ∫2−3sin⁡(u)cos⁡(u)4 du=∫2−3sin⁡(u)cos⁡(u) du4\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=2−3sin⁡(u)u = 2^{- 3 \sin{\left(u \right)}}.

          Then let du=−3⋅2−3sin⁡(u)log⁡(2)cos⁡(u)dudu = - 3 \cdot 2^{- 3 \sin{\left(u \right)}} \log{\left(2 \right)} \cos{\left(u \right)} du and substitute −du3log⁡(2)- \frac{du}{3 \log{\left(2 \right)}}:

          ∫19log⁡(2)2 du\int \frac{1}{9 \log{\left(2 \right)}^{2}}\, du

          1. The integral of a constant times a function is the constant times the integral of the function:

            ∫(−13log⁡(2)) du=−∫1 du3log⁡(2)\int \left(- \frac{1}{3 \log{\left(2 \right)}}\right)\, du = - \frac{\int 1\, du}{3 \log{\left(2 \right)}}

            1. The integral of a constant is the constant times the variable of integration:

              ∫1 du=u\int 1\, du = u

            So, the result is: −u3log⁡(2)- \frac{u}{3 \log{\left(2 \right)}}

          Now substitute uu back in:

          −2−3sin⁡(u)3log⁡(2)- \frac{2^{- 3 \sin{\left(u \right)}}}{3 \log{\left(2 \right)}}

        So, the result is: −2−3sin⁡(u)12log⁡(2)- \frac{2^{- 3 \sin{\left(u \right)}}}{12 \log{\left(2 \right)}}

      Now substitute uu back in:

      −2−3sin⁡(4x)12log⁡(2)- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}

    Method #2

    1. Let u=123sin⁡(4x)u = \frac{1}{2^{3 \sin{\left(4 x \right)}}}.

      Then let du=−12⋅2−3sin⁡(4x)log⁡(2)cos⁡(4x)dxdu = - 12 \cdot 2^{- 3 \sin{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(4 x \right)} dx and substitute −du12log⁡(2)- \frac{du}{12 \log{\left(2 \right)}}:

      ∫1144log⁡(2)2 du\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:

        ∫(−112log⁡(2)) du=−∫1 du12log⁡(2)\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:

          ∫1 du=u\int 1\, du = u

        So, the result is: −u12log⁡(2)- \frac{u}{12 \log{\left(2 \right)}}

      Now substitute uu back in:

      −2−3sin⁡(4x)12log⁡(2)- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}

    Method #3

    1. Let u=23sin⁡(4x)u = 2^{3 \sin{\left(4 x \right)}}.

      Then let du=12⋅23sin⁡(4x)log⁡(2)cos⁡(4x)dxdu = 12 \cdot 2^{3 \sin{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(4 x \right)} dx and substitute du12log⁡(2)\frac{du}{12 \log{\left(2 \right)}}:

      ∫1144u2log⁡(2)2 du\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:

        ∫112u2log⁡(2) du=∫1u2 du12log⁡(2)\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 unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫1u2 du=−1u\int \frac{1}{u^{2}}\, du = - \frac{1}{u}

        So, the result is: −112ulog⁡(2)- \frac{1}{12 u \log{\left(2 \right)}}

      Now substitute uu back in:

      −2−3sin⁡(4x)12log⁡(2)- \frac{2^{- 3 \sin{\left(4 x \right)}}}{12 \log{\left(2 \right)}}

  2. Add the constant of integration:

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

The answer is:

−2−3sin⁡(4x)12log⁡(2)+constant- \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                                
 |                                  
/
∫cos⁡(4x)23sin⁡(4x) dx=C−2−3sin⁡(4x)12log⁡(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
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
             -3*sin(4)
    1       2         
--------- - ----------
12*log(2)   12*log(2)
−2−3sin⁡(4)12log⁡(2)+112log⁡(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)
−2−3sin⁡(4)12log⁡(2)+112log⁡(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
Integral of cos(4x)/2^(3*sin(4x)) dx

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

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