Integral of sin(2x/3) dx

The solution

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 3*pi           
   /            
  |             
  |     /2*x\   
  |  sin|---| dx
  |     \ 3 /   
  |             
 /              
 0

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

Integral(sin((2*x)/3), (x, 0, 3*pi))

Detail solution

Let $u = \frac{2 x}{3}$ .

Then let $du = \frac{2 dx}{3}$ and substitute $\frac{3 du}{2}$ :

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

$- \frac{3 \cos{\left(\frac{2 x}{3} \right)}}{2}$
Now simplify:

$- \frac{3 \cos{\left(\frac{2 x}{3} \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                       /2*x\
 |                   3*cos|---|
 |    /2*x\               \ 3 /
 | sin|---| dx = C - ----------
 |    \ 3 /              2     
 |                             
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

6.11036069524454e-22

6.11036069524454e-22

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

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Integral of sin(2x/3) dx

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