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Identical expressions
four *cos(two *x)* five ^sin(two *x)
4 multiply by co sinus of e of (2 multiply by x) multiply by 5 to the power of sinus of (2 multiply by x)
four multiply by co sinus of e of (two multiply by x) multiply by five to the power of sinus of (two multiply by x)
4*cos(2*x)*5sin(2*x)
4*cos2*x*5sin2*x
4cos(2x)5^sin(2x)
4cos(2x)5sin(2x)
4cos2x5sin2x
4cos2x5^sin2x
4*cos(2*x)*5^sin(2*x)dx

Solving integrals/ 4*cos(2*x)*5^sin(2*x)

Integral of 4cos(2x)5^sin(2x) dx

The solution

You have entered [src]

  1                        
  /                        
 |                         
 |              sin(2*x)   
 |  4*cos(2*x)*5         dx
 |                         
/                          
0

$$\int\limits_{0}^{1} 5^{\sin{\left(2 x \right)}} 4 \cos{\left(2 x \right)}\, dx$$

Integral((4*cos(2*x))*5^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 $2 du$ :
  
  $\int 2 \cdot 5^{\sin{\left(u \right)}} \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 5^{\sin{\left(u \right)}} \cos{\left(u \right)}\, du = 2 \int 5^{\sin{\left(u \right)}} \cos{\left(u \right)}\, du$
    1. Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int 5^{u}\, du$
      
      The integral of an exponential function is itself divided by the natural logarithm of the base.
      
      $\int 5^{u}\, du = \frac{5^{u}}{\log{\left(5 \right)}}$
      
      Now substitute $u$ back in:
      
      $\frac{5^{\sin{\left(u \right)}}}{\log{\left(5 \right)}}$
    So, the result is: $\frac{2 \cdot 5^{\sin{\left(u \right)}}}{\log{\left(5 \right)}}$
  Now substitute $u$ back in:
  
  $\frac{2 \cdot 5^{\sin{\left(2 x \right)}}}{\log{\left(5 \right)}}$
Method #2
1. Let $u = \sin{\left(2 x \right)}$ .
  
  Then let $du = 2 \cos{\left(2 x \right)} dx$ and substitute $2 du$ :
  
  $\int 2 \cdot 5^{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 5^{u}\, du = 2 \int 5^{u}\, du$
    1. The integral of an exponential function is itself divided by the natural logarithm of the base.
      
      $\int 5^{u}\, du = \frac{5^{u}}{\log{\left(5 \right)}}$
    So, the result is: $\frac{2 \cdot 5^{u}}{\log{\left(5 \right)}}$
  Now substitute $u$ back in:
  
  $\frac{2 \cdot 5^{\sin{\left(2 x \right)}}}{\log{\left(5 \right)}}$
Add the constant of integration:

$\frac{2 \cdot 5^{\sin{\left(2 x \right)}}}{\log{\left(5 \right)}}+ \mathrm{constant}$

The answer is:

$\frac{2 \cdot 5^{\sin{\left(2 x \right)}}}{\log{\left(5 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                  sin(2*x)
 |             sin(2*x)          2*5        
 | 4*cos(2*x)*5         dx = C + -----------
 |                                  log(5)  
/

$$\int 5^{\sin{\left(2 x \right)}} 4 \cos{\left(2 x \right)}\, dx = \frac{2 \cdot 5^{\sin{\left(2 x \right)}}}{\log{\left(5 \right)}} + C$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

              sin(2)
    2      2*5      
- ------ + ---------
  log(5)     log(5)

$$- \frac{2}{\log{\left(5 \right)}} + \frac{2 \cdot 5^{\sin{\left(2 \right)}}}{\log{\left(5 \right)}}$$

              sin(2)
    2      2*5      
- ------ + ---------
  log(5)     log(5)

$$- \frac{2}{\log{\left(5 \right)}} + \frac{2 \cdot 5^{\sin{\left(2 \right)}}}{\log{\left(5 \right)}}$$

-2/log(5) + 2*5^sin(2)/log(5)

Numerical answer [src]

4.12675035857694

4.12675035857694

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

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How to use it?

Identical expressions

Integral of 4cos(2x)5^sin(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Integral of 4*cos(2*x)*5^sin(2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of 4cos(2x)5^sin(2x) dx