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sin(6x-П/ three)
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Similar expressions
sin(6x+П/3)

Integral of sin(6x-П/3) dx

The solution

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  1                 
  /                 
 |                  
 |     /      pi\   
 |  sin|6*x - --| dx
 |     \      3 /   
 |                  
/                   
0

$$\int\limits_{0}^{1} \sin{\left(6 x - \frac{\pi}{3} \right)}\, dx$$

Integral(sin(6*x - pi/3), (x, 0, 1))

Detail solution

Let $u = 6 x - \frac{\pi}{3}$ .

Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :

$\int \frac{\sin{\left(u \right)}}{6}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{6}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{6}$
Now substitute $u$ back in:

$- \frac{\cos{\left(6 x - \frac{\pi}{3} \right)}}{6}$
Now simplify:

$- \frac{\sin{\left(6 x + \frac{\pi}{6} \right)}}{6}$
Add the constant of integration:

$- \frac{\sin{\left(6 x + \frac{\pi}{6} \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{\sin{\left(6 x + \frac{\pi}{6} \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          /      pi\
 |                        cos|6*x - --|
 |    /      pi\             \      3 /
 | sin|6*x - --| dx = C - -------------
 |    \      3 /                6      
 |                                     
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

        /    pi\
     sin|6 + --|
1       \    6 /
-- - -----------
12        6

$$\frac{1}{12} - \frac{\sin{\left(\frac{\pi}{6} + 6 \right)}}{6}$$

        /    pi\
     sin|6 + --|
1       \    6 /
-- - -----------
12        6

$$\frac{1}{12} - \frac{\sin{\left(\frac{\pi}{6} + 6 \right)}}{6}$$

1/12 - sin(6 + pi/6)/6

Numerical answer [src]

0.043649296054362

0.043649296054362

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

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Integral of sin(6x-П/3) dx

Limits of integration:

The graph:

Piecewise:

The solution