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Similar expressions
(2/3)(sen(4x)dx)

Integral of (-2/3)(sen(4x)dx) dx

The solution

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

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

Integral(-2*sin(4*x)/3, (x, 0, 1))

Detail solution

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

$\int \left(- \frac{2 \sin{\left(4 x \right)}}{3}\right)\, dx = - \frac{2 \int \sin{\left(4 x \right)}\, dx}{3}$
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\sin{\left(u \right)}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(4 x \right)}}{4}$
So, the result is: $\frac{\cos{\left(4 x \right)}}{6}$
Add the constant of integration:

$\frac{\cos{\left(4 x \right)}}{6}+ \mathrm{constant}$

The answer is:

$\frac{\cos{\left(4 x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             
 |                              
 | -2*sin(4*x)          cos(4*x)
 | ----------- dx = C + --------
 |      3                  6    
 |                              
/

$$\int \left(- \frac{2 \sin{\left(4 x \right)}}{3}\right)\, dx = C + \frac{\cos{\left(4 x \right)}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   cos(4)
- - + ------
  6     6

$$- \frac{1}{6} + \frac{\cos{\left(4 \right)}}{6}$$

  1   cos(4)
- - + ------
  6     6

$$- \frac{1}{6} + \frac{\cos{\left(4 \right)}}{6}$$

-1/6 + cos(4)/6

Numerical answer [src]

-0.275607270143935

-0.275607270143935

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

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Integral of (-2/3)(sen(4x)dx) dx

Limits of integration:

The graph:

Piecewise:

The solution