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Derivative of:
sin(2*x+pi/3)
Identical expressions
sin(two *x+pi/ three)
sinus of (2 multiply by x plus Pi divide by 3)
sinus of (two multiply by x plus Pi divide by three)
sin(2x+pi/3)
sin2x+pi/3
sin(2*x+pi divide by 3)
sin(2*x+pi/3)dx
Similar expressions
sin(2*x-pi/3)

Integral of sin(2*x+pi/3) dx

The solution

You have entered [src]

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

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

Integral(sin(2*x + pi/3), (x, -pi, pi))

Detail solution

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

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

-2.12519032545733e-16

-2.12519032545733e-16

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

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Identical expressions

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Integral of sin(2*x+pi/3) dx

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The graph:

Piecewise:

The solution