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Identical expressions
two /3sin(3x)
2 divide by 3 sinus of (3x)
two divide by 3 sinus of (3x)
2/3sin3x
2 divide by 3sin(3x)
2/3sin(3x)dx

Integral of 2/3sin(3x) dx

The solution

You have entered [src]

 pi              
 --              
 4               
  /              
 |               
 |  2*sin(3*x)   
 |  ---------- dx
 |      3        
 |               
/                
pi               
--               
6

$$\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{2 \sin{\left(3 x \right)}}{3}\, dx$$

Integral(2*sin(3*x)/3, (x, pi/6, pi/4))

Detail solution

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

$\int \frac{2 \sin{\left(3 x \right)}}{3}\, dx = \frac{2 \int \sin{\left(3 x \right)}\, dx}{3}$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\sin{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(3 x \right)}}{3}$
So, the result is: $- \frac{2 \cos{\left(3 x \right)}}{9}$
Add the constant of integration:

$- \frac{2 \cos{\left(3 x \right)}}{9}+ \mathrm{constant}$

The answer is:

$- \frac{2 \cos{\left(3 x \right)}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                               
 | 2*sin(3*x)          2*cos(3*x)
 | ---------- dx = C - ----------
 |     3                   9     
 |                               
/

$$\int \frac{2 \sin{\left(3 x \right)}}{3}\, dx = C - \frac{2 \cos{\left(3 x \right)}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ___
\/ 2 
-----
  9

$$\frac{\sqrt{2}}{9}$$

  ___
\/ 2 
-----
  9

$$\frac{\sqrt{2}}{9}$$

sqrt(2)/9

Numerical answer [src]

0.157134840263677

0.157134840263677

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

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Identical expressions

Integral of 2/3sin(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution